by : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA

by : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA
Real number by : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA

Isi dengan Judul Halaman Terkait
Competency standard : Solve problems related to the concept of operating the real. Hal.: 2 Isi dengan Judul Halaman Terkait

Learning Materials : System Real Numbers On integer operations Operations on Numbers fraction Conversion Numbers Comparison, scale and Percent Implementation of Real Numbers In completing the Program Expertise . Hal.: 3 Isi dengan Judul Halaman Terkait

Applying the operation on the real . Isi dengan Judul Halaman Terkait
basic competence 1 : Applying the operation on the real . Hal.: 4 Isi dengan Judul Halaman Terkait

1. System real number Learning Objectives : Students can: Distinguish the various real Hal.: 5 Isi dengan Judul Halaman Terkait

Scheme real number : Real number Rational number Irrational number Fraction number Integer 0(zero) Positive integer (natural number) Negative integer Composite number Prime number 1 Finish Hal.: 6 Isi dengan Judul Halaman Terkait

Rational numbers Rational number is number that can be expressed with the form where a and b integer . example : 5/6, 7/8 ,-3/4, etc. Hal.: 7 Isi dengan Judul Halaman Terkait

Irrational number is number that can not be expressed with the form example : 5, 7, ∏, etc. Hal.: 8 Isi dengan Judul Halaman Terkait

2. integer operations Learning objectives : Students can : 1. Complete the operation Answer and integer . 2. Complete the operation multiplication and division of integer . Hal.: 9 Isi dengan Judul Halaman Terkait

a. Addition and subtraction addition integer can equipment with equipment or the movement along the track represents the number lines. First on the positive integer and zero, and then to develop a rounded whole. One model of learning to use the steps in the line is to make agreements, for example: Addition : Positive, advanced Negative, back Subtraction : Positive, advanced Hal.: 10 Isi dengan Judul Halaman Terkait

= .... step 3 -1 1 2 3 4 5 6 7 -2 -3 -4 -5 -6 -7 -2 So that the results of the additional : = 1 Hal.: 11 Isi dengan Judul Halaman Terkait

5 + (-2) = .... step 5 -1 1 2 3 4 5 6 7 -2 -3 -4 -5 -6 -7 2 So that the results of the additional : 5 + (-2) = 3 Hal.: 12 Isi dengan Judul Halaman Terkait

-2 - (-5) = .... step -2 -1 1 2 3 4 5 6 7 -2 -3 -4 -5 -6 -7 -5 So that the results of the additional : -2 - (-5) = 3 etc Hal.: 13 Isi dengan Judul Halaman Terkait

b. Integer multiplication Multiplication and division of integer is the development of multiplication and division of the original. Which is still a problem is often a sign of problems in the operation. Can be explained with the pattern, for example, how to fill the box. . . in the table below:  –3 –2 – – So that will be obtained: Results marked with the same time the number of positive results and if the sign is negative the result is different, and each number (rounded) 0 multiplied by the result 0 (zero). Hal.: 14 Isi dengan Judul Halaman Terkait

c. Integer division With a distribution of b is declared with a: b or a: b = c if and only if a = b x c. Implications, among other : 1). For each b  0, 0 : b = 0 because 0 x b = 0  0 = 0  b correct 2). c x 0  0 = 0  c. because 0  c = 0 for each c, then c how can any value, including negative numbers, the results of c : 0 said not a . 3). For each a  0, a : 0 = c may not happen, because there is no value c that meet the a = c  0, or frequently stated ∞ the result is not defined. Hal.: 15 Isi dengan Judul Halaman Terkait

3. Fraction number Initial concept of a fraction is part of the whole (with the buckle geometry) instance : How the operation fraction number ? instance : that this result can equipment Hal.: 16 Isi dengan Judul Halaman Terkait

exercise Ali has 2 meter wire, will be given to friends to create interest. If each person gets the 2 / 5 meters, how many people get that part? Of 2 ¾ kg sugar akan made bread recipe. If a recipe requires sugar as much as ½ kg. How many bread recipes that can be made? If a job is done by 4 people will finish in 30 days. If done by the usual 6 will be completed within 80 days. How much time needed to complete the work if it is done jointly by the 3 and 4 experienced people? Hal.: 17 Isi dengan Judul Halaman Terkait

4. The fractional conversion
Change the Ordinary to the fraction or a decimal fraction . Change the Ordinary to the form of a fraction or vice versa Percent . Fractional change to Decimal Percent of, or otherwise . Hal.: 18 Isi dengan Judul Halaman Terkait

Change the fraction to fraction Ordinary Decimal
Change the Ordinary to fraction decimal fraction, or vice versa. Example : = ….. Do with the usual . 0,7 = 7/10 0,08 = 8/100 = 2/25, etc. Hal.: 19 Isi dengan Judul Halaman Terkait

Change the fraction to the percent of
To change the fraction to a normal form of percent, we multiply the fraction by 100% . Meanwhile, to change the percent into a fraction of normal, we must divide the number by 100 percent . Hal.: 20 Isi dengan Judul Halaman Terkait

example 4/5 = …… % = 4/5 x 100 % = 80 % 2/3 = …… % = 2/3 x 100 % = 66 2/3 % 15 % = …… = 15/100 = 3/20 etc. Hal.: 21 Isi dengan Judul Halaman Terkait

Change the fraction to a decimal form of a percent
To change a decimal to fraction of percent, we multiply the fraction by 100%. Meanwhile, to change the form of a percent into a decimal fraction, we must divide the number by 100 percent. Hal.: 22 Isi dengan Judul Halaman Terkait

example 0,03 = ……% = 0,03 x 100% = 3% 0,056 = ……% = 0,056 x 100% = 5,6% 35% = ……. (decimal fraction ) = 35/100 = 0,35 425% = ……… = 425/100 = 4,25 etc. Hal.: 23 Isi dengan Judul Halaman Terkait

exercise 1. Indicate the following fraction into the form of a decimal and percent : a. 3/ c. 2/3 (three decimal places ) b. 7/ d. 5/6 (three decimal places ) 2. Specify the form of a percent below the fraction and decimal fraction normal : a. 35% c. 12 ½ % b. 8 % d. 16 2/3 % Hal.: 24 Isi dengan Judul Halaman Terkait

5. ratio There are two kinds of comparisons, namely: Comparison worth . Comparison Value turn . Try our eye on each comparison is . Hal.: 25 Isi dengan Judul Halaman Terkait

Kind of comparison Worth Turn value Many (Fruit) Price (Rupiah) 1 2 3 4 … 6 x 200 400 Speed. (Km / hr) Time (Hours) 60 30 20 … 5 x 1 2 Hal.: 26 Isi dengan Judul Halaman Terkait

Kind of comparison Comparison worth a = k  b, k  R when b then a, proportionate increase in straight when b then a , proportionate increase in straight instance : s = v  t Comparison Value turn when b then a  when b  then a  instance : v = s/t Hal.: 27 Isi dengan Judul Halaman Terkait Misal :

Comparison of advanced
Learning Experience With 24 workers, a job is complete bulk sewing planned within 48 days. After working for 12 days with 24 workers, stopped work for 9 days because of it. How many workers who should be added so that work can be completed on time? Hal.: 28 Isi dengan Judul Halaman Terkait

Concerning the settlement of the above :
Turn the value of comparison, so that: The remaining work to 48–12 = 36 day which should be completed by 24 people. But the only remaining 48–12–9 = 27 day. So obtained: 24 people  day x people  day then: x = 36/27 x 24 = 32 So the additional 8 workers Hal.: 29 Isi dengan Judul Halaman Terkait

example With fixed velocity, a car requires 5 liters of petrol to 60 km distance. How many liters of gasoline needed to travel 150 km distance? The distance between two cities can be a vehicle with an average speed of 72 km / hr for 5 hours. What is the average speed of vehicles to travel long distances if the 8 hours? Hal.: 30 Isi dengan Judul Halaman Terkait

solution: Because comparison worth then : 150/60 x 5 km = 12,5 liters. Comparison value turn, then : 5/8 x 72 km/hr = 45 km/hr. Hal.: 31 Isi dengan Judul Halaman Terkait

exercise Making cake mix liquids consisting of coconut oil and water with the comparison 1: 18. How many liters of coconut oil is needed to obtain 9.5 liters of fluid mixtures? A map of the oblong drawn with scale: 1: 120,000 and has a length: width is 4:3. Meanwhile, around 112 cm map. actually a broad set described by this map? KE MENU AWAL Hal.: 32 Isi dengan Judul Halaman Terkait

Basic competence 2 : Applying the operation have an important position on the number. Hal.: 33 Isi dengan Judul Halaman Terkait

learning materials : Concepts have an important position and the nature-nature. Numbers have an important position in the operation. The simplification have an important position. Hal.: 34 Isi dengan Judul Halaman Terkait

exponent Ex: IN PROCESS AMUBA cleavage SEL Hal.: 35 Isi dengan Judul Halaman Terkait

Learning objectives Explain the definition of rank Explain the nature have an important position Have an important position to operate the appropriate concept unanimously positive. Hal.: 36 Isi dengan Judul Halaman Terkait

The designation ON REAL Numbers
If a number is real and n is a positive integer, then an is defined as a multiple of n times, written a = a x a x a x x a n factor with, a is a form of public have an important position of the , a called the principal or base , a R, n called the rank of (exponent) n n Hal.: 37 Isi dengan Judul Halaman Terkait

start Looking exponent = looking logarithm 2... = 2048 2log 2048 = ... ? after … PERIOD 1 2 3 4 5 6 … 9 11 many amuba 512 ? 1 2 4 8 16 32 64 ? … 2048 2 2 2 2 2  2 2  2  2 2  2  2  2 Numbers have an important position notation 211 2? 2? 2? 20 21 22 23 24 25 2... 26 2? 29 2? 2? Hal.: 38 Isi dengan Judul Halaman Terkait

definition : a1 = a Are all the results obtained form of rational number? What rational number with the rank of integer numbers is always rational? Whether the division of two rational number (not a divider 0) always produce rational number? How does nature have an important position the operation? Hal.: 39 Isi dengan Judul Halaman Terkait

Consider the following examples : = 2 x 2 x 2 x 2 x 2 x 2 x 2, 7 factor (-3) = (-3) x (-3) x (-3) x (-3) x (-3) 5 factor 8 factor In the example above, 2, -3, and is a number (base), while 7, 5, and 8 is the rank of (exponent). 7 5 Hal.: 40 Isi dengan Judul Halaman Terkait

The attributes have an important position : Attributes have an important position with the designation unanimously positive is as follows. Suppose a, b R, and m, n is a positive integer, then S-1 S-2 S-3 S-4 S-5 Hal.: 41 Isi dengan Judul Halaman Terkait

With experiment :  2  2  2  2  2 2  2  2  2  2 4 factor number 2 6 factor number 2 Means 24+6 = 210 (4 + 6) factor number 2  k  k  k k  k  k  k  k  k  k  k  k 3 factor number k 9 factor number k Means k3+9 = k12  (3 + 9) factor number k a  a  a  …  a  a a  a  …  a q factor number a p factor number a  ap  aq = ap+q (p + q) factor number a Means ap+q Contoh: x5  x 12= x5+12 = x17 32  33 = = 35 76  713= = 319 Hal.: 42 Isi dengan Judul Halaman Terkait

With experiment: ? 2  2  2 2  2  2  2 __________ result ( = ) 2  2  2 = 23 = 27 : 24 = 2  2  2  2 Means 27 : 24 = p factor number a a  a  a  a  a  a …  a a x a x a x … x a __________ ap : aq = (p >q) = a  a  a...  a a a  a … a p – q factor q factor number a = ap-q Means  ap : aq = ap ‑ q Example: 36 : 34 = 32 = 713-8 = 75 713 : 78= Hal.: 43 Isi dengan Judul Halaman Terkait

With experiment: p factors p number of factors a ap a  a  a  a  a  a …  a _______________________ ____ = = b  b  b  b  b  b …  b bp p number of factors b ap ____ therefore : bp Hal.: 44 Isi dengan Judul Halaman Terkait

exercise Write operation results in the following form of the most simple: = …… 36 2. 53 x 23 = …… = …….. 23 Hal.: 45 Isi dengan Judul Halaman Terkait

Basic competence 3 : Applying the operation on the Irrasional. Hal.: 46 Isi dengan Judul Halaman Terkait

Learning materials : The concept of the Irrasional Production Numbers In the form of roots. Simplification of the root. Hal.: 47 Isi dengan Judul Halaman Terkait

Learning objectives Explain the definition irrasional Operate a number of roots. Simplify the number irrasional. Hal.: 48 Isi dengan Judul Halaman Terkait

Root ( apersepsi ) The Square Root Extent of a garden 225 m2. If the garden is shaped square, the length of the garden? Square footage is 15 m, from because 152 = 225. Root of the withdrawal process For each a  0 and b  0, if and only if a2 = b Why b can not be negative? Hal.: 49 Isi dengan Judul Halaman Terkait

Learning experience mr Karyo have sebidang the extent of land recorded in the Office land is 1369 m2. In fact the form of a square yard. How long is the size of the ground yards? How many meters of iron rod to create the necessary framework of iron below 6 m 12 m Hal.: 50 Isi dengan Judul Halaman Terkait

how the value of : Settlement: Which is true! Hal.: 51 Isi dengan Judul Halaman Terkait

example: how the value of 1369 m2 berapa meter? 900 m2 30 m h m (2) (1) (3) h m (1) (2) (3) (2  30 + h)m then: 2  30h + h2 = 1369 – 900 (2  30 + ….) …. = 469 Retrieved :h=7 So the result 37 Hal.: 52 Isi dengan Judul Halaman Terkait

method can be written : = 302 = 469 (2  30 + …)  … = 469 (–) Hal.: 53 Isi dengan Judul Halaman Terkait

Cube root 8 m3 ? = p Water tank inside the cuboid can accommodate as many as 8 m3 of water. Size of the bathtub? Cistern size is 2 m, of p  p  p = 8  p = 2 because 23 = 8. If and only if b3 = a Hal.: 54 Isi dengan Judul Halaman Terkait

= ? 3 example, how the value of : 6 (p + s)3 = p3 + 3p2s + 3ps2 + s3 = ? 3 = p3 + (3p2 + 3ps + s2 )s 3  202 203 = 3  20 9 5 7 6 9 5 7 6 (1200 + 60 ... 6 )  ... 6 6 ( )  6 1596  6 = 26 3 so: Hal.: 55 Isi dengan Judul Halaman Terkait

Root operation Basic Operation : For a  0 and b  0 n  A, n  2, = a, from real. Answer and Reduction can be simplified if a similar radical example : = = ( 5 – ) = 2 Multiple forms of Roots with the nature The form of the distribution of roots Hal.: 56 Isi dengan Judul Halaman Terkait

root Rationalizing the denominator of a fraction Forms: or multiplying forms:(ab + cd) where a, b, c, and d rasional, and at least one of the b and d irrational x = Hal.: 57 Isi dengan Judul Halaman Terkait

Basic competence 4 : Applying the concept of logarithm Hal.: 58 Isi dengan Judul Halaman Terkait

Learning materials : The concept of logarithm The logarithm operation. Hal.: 59 Isi dengan Judul Halaman Terkait

Learning objectives Explain the concept of logarithm Explain the nature logarithm. Logarithm operation with nature logarithm. Problem solving program expertise relating to the logarithm. Hal.: 60 Isi dengan Judul Halaman Terkait

LOGARITHM In the form of 2log 128 In the form of alog b 2 is the base logarithm b is the base logarithm 2log 128 = 7  27 = 128 alog b = c  ac = b In general it was stated that : c replaced alog b c = alog b Appropriate number of discussion have an important position, then a > 0, b > 0 and a  1, because 1log b = c  1c = b, for b  1 never found value of c. alog b = c alog b = c alog b = c alog b = c  ac = b alog b a = b logarithm of b to the base a ditulis alog b is number if a exponent number result of b Hal.: 61 Isi dengan Judul Halaman Terkait

LOGARITHM Note : ab = c ab = …. search results promotion …b = c find square root of b c a... = c find that designation of a result c = find that designation of a result c = alog c = … alog b = c  ac = b with a > 0 , a  1 and b > 0 Hal.: 62 Isi dengan Judul Halaman Terkait

The properties of logarithm :
if a > 0 , a  1 , m > 0 , n > 0 and x  R, then : alog ax = x alog (m.n) = alog m + alog n alog (m/n) = alog m - alog n alog mx = x. alog m alog m = if g > 0 , g  1 etc. Hal.: 63 Isi dengan Judul Halaman Terkait

exercise: 1. Simple form of 2log 4 + 2log 12 – 2log 6 is ….. a b c d e. 8 2. If known 3log 2 = p, then 9log 8 is... . 3. if 5log 2 = p then log 2,5 = … Hal.: 64 Isi dengan Judul Halaman Terkait

exercise: An optical equipment needed to observe the stars at the top to a height of six, the limit of naked eye view. However, the tool has limitations. L limit the height of a telescope with optical inch in diamater D formulated in L = log D Look for the height limit for the telescope similar to the 6 inch diameter . Find the diameter of a lens that has a height limit of 20.6. Hal.: 65 Isi dengan Judul Halaman Terkait

By : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA
MATRIX By : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA

Competency standards :
Solve problems related to the concept of matrix. Hal.: 67 Isi dengan Judul Halaman Terkait

Learning materials: Kind of Matrix. Matrix operation. Hal.: 68 Isi dengan Judul Halaman Terkait

Describe your kind of matrix. Isi dengan Judul Halaman Terkait
Basic competence 1 : Describe your kind of matrix. Hal.: 69 Isi dengan Judul Halaman Terkait

Learning objectives Explaining the matrix Explain the notation and matrix-element. Distinguish the type of matrix. Complete similarity matrix. Explaining Transpose Matrix. Hal.: 70 Isi dengan Judul Halaman Terkait

Matrix Matrix is the order of the numbers-which consists of these lines and columns. Each number in the matrix is called the entry or elements. Ordo (size) of the matrix is the number of rows times the number of columns. a a12…….a1j ……a1n a a22 ……a2j…….a2n : : : : ai ai2 ……aij…….. ain : : : : am1 am2……amj……. amn Matriks adalah array (susunan) bilangan-bilangan yang terdiri atas baris-baris dan kolom-kolom. Masing-masing bilangan dalam matriks disebut entri atau elemen. Ordo (ukuran) matriks adalah jumlah baris kali jumlah kolom. Matriks persegi adalah matriks yang jumalah baris sama dengan jumlah kolom. Perhatikan dengan baik notasi yang dipakai. Notasi ini akan dipergunakan terus dalam pemelajaran selanjutnya. Notation: Matrix: A = [aij] Element: (A)ij = aij Ordo A: m x n A = row column Hal.: 71 Isi dengan Judul Halaman Terkait

General form Matrix A = Hal.: 72 Isi dengan Judul Halaman Terkait

example: Ordo matrix P = 3 x 3 P = 1 Entry that is located on line 1 column 2 is 6 Entry that is located on the first row third column is Members of the matrix P in row 3 = Members of the matrix P in column 1 = Hal.: 73 Isi dengan Judul Halaman Terkait

SQUARE MATRIKS Matrix is a square matrix is the number of rows and columns the same number of. Trace(A) = Matriks persegi (bujur sangkar) adalah matriks yang jumlah baris dan jumlah kolom sama. Matriks persegi mempunyai dua diagonal, diagonal yang ke kanan disebut diagonal utama. Pada matriks persegi didefinisikan trace. Trace dari matriks adalah jumlahan elemen-elemen diagonal utama Main diagonal Trace from matriks is sum the diagonal elements of the main Hal.: 74 Isi dengan Judul Halaman Terkait

Zero matrix and identity
zero matrix is matrix of all elements zero 0 0 0 0 identity matrix is a square matrix of elements of main diagonal elements 1 and other 0 I3 I4 I2 Matriks nol dan identitas Matriks nol adalah matriks yang setiap entrinya nol. Matriks nol adalah matriks identitas terhadap jumlahan, artinya jika Dijumlahkan dengan matriks lain A yang berukuran sama maka matriks A tersebut tidak berubah. Matriks identitas I adalah matrks persegi yang entri diagonal utamanya satu dan entri lainnya nol. Matriks identitas merupakan elemen identitas terhadap perkalian, sebab, jika dikaliakan dengan matriks lain A Yang berukran sama dengan I maka hasilnya sama dengan A. Matriks nol disebut matrik identitas terhadap jumlahan, matriks identitas disebut matriks identitas terhadap perkalian matriks Hal.: 75 Isi dengan Judul Halaman Terkait

Transpose A = AT = A’ = Definition: Transpose the matrix A is the matrix AT-columns are rows of the A-line, lines is from the columns of A . Transpose matriks Transpose dari matriks adalah matriks baru yang kolom –kolom menjadi baris-baris. Matriks simetri adalah matriks yang sama dengan transposenya. Matriks orthogonal adalah matriks yang inversenya sama dengan trnsposenya. Transpose dari transpose A adalaha A. Jumlahan dua matriks sama dengan jumlahan transposenya, transpose dari skalar kali matriks sama dengan skalar kali trenasposenya. Transpose dari hasil kali sama dengan hasil kali transposenya dengan urutan terbalik. Tampilkan contoh matriks 2x4, kmd transposekan Berikan definisi umum dari transpose. Matriks A simetri jkk A = AT, berikan contoh Matriks A orthogonal jika dan hanya jika AT = A –1, berikan contoh: matriks rotasi. Berikan contoh dan rumus umum (A-1)T = (AT)-1 [AT]ij = [A]ji n x m If A is m x n matrix, the matrix transpose AT size ……….. Hal.: 76 Isi dengan Judul Halaman Terkait

Properties matrix transpose
Transpose from matrix A transpose is A: (AT )T = A A (AT)T = A AT example: Transpose dari transpose adalah matriks iu sendiri Hal.: 77 Isi dengan Judul Halaman Terkait

Properties matrix transpose 2. (A+B)T = AT + BT A T AT BT B T A+B (A+B)T T = + 2.Transpose dari jumlaham nmatriks sama dengan jumlahan transpose-transposenya Hal.: 78 Isi dengan Judul Halaman Terkait

Properties matrix transpose 3. (kA)T = k(A) T from scalar k kA T A T k 3. Transpose dari hasil kali skalr dengan matriks sama dengan hasil kali skalar dengan transposenya (kA)T = k(A)T Hal.: 79 Isi dengan Judul Halaman Terkait

Properties matrix transpose 4. (AB)T = BT AT B T T A AB T = 4. Transpose hasil kali A dan B sama dengan hasil kali transpose a dan transpose B. (AB)T AB = BTAT Hal.: 80 Isi dengan Judul Halaman Terkait

matrix symmetry Matrix A is symmetry if and only if A = AT A’ = A = A symmetry Matriks simetri adalah matriks yang sama dengan transposenya. Matriks ortogonal adalah matriks yang inversenya sama dengan transposenya. = AT A = Hal.: 81 Isi dengan Judul Halaman Terkait

orthogonal matrix Matrix A orthogonal if and only if AT = A –1 0 1 AT= A = = A-1 BT= ½√2 ½√2 -½√2 ½√2 B = ½√2 -½√2 ½√2 ½√2 = B-1 Matriks ortogonal adalah matriks yang inversenya sama dengan transposenya. A AT (A-1)T = (AT)-1 if A is orthogonal matrix, then (A-1)T = (AT)-1 Hal.: 82 Isi dengan Judul Halaman Terkait

similarity 2 matrix & summation 2 matrix Hal.: 83 Isi dengan Judul Halaman Terkait

similarity 2 matrix condition : * have the same ordo * The same entry and the same location for each i and j, aij = bij Hal.: 84 Isi dengan Judul Halaman Terkait

Similarity 2 matrix Two matrix with the same size and if each entry is associated same . B = A = A = B D = C = C ≠ D x F = E = F if x = 1 E = Dua matriks sama jika dan hanya jika ukurannya sama dan setiap entri yang bersesuaian sama. Contoh, A = B sebab setiap entri yangyang letaknya bersesuaian adalah sama. C tidak sama dengan D sebab entri baris pertama kolom ke dua matriks C dan D tidak sama. E dan F sama hanya jika x = 1, selai itu, E tidak sama dengan F. Apakah G sama dengan H? H = ? ? ? 2 2 5 6 9 0 7 G = 2 2 2 G = H 4 5 6 9 7 Hal.: 85 Isi dengan Judul Halaman Terkait

example: find the values , , and If given matrix A = matrix B! Hal.: 86 Isi dengan Judul Halaman Terkait

Example (next): - 5 = A= = -1 = 5 B= Hal.: 87 Isi dengan Judul Halaman Terkait

Basic competence 2 : Complete the operation matrix Hal.: 88 Isi dengan Judul Halaman Terkait

Learning objectives Explain the operation matrix. Complete the operation summation and reduction matrix. Complete the matrix multiplication operation with scalar. Complete the two matrix multiplication operations. Determine the inverse matrix square ordo two. Complete SPL with matrix. Hal.: 89 Isi dengan Judul Halaman Terkait

summation 2 matrix Procedure: Enumerate each entry / element A matrix with each entry / element matrix B which seletak (rows and columns the same) Hal.: 90 Isi dengan Judul Halaman Terkait

Summation and reduction two matrix
example A = B = A + B = = 10+2 22+6 12 28 1+7 -1+5 8 4 A - B = = Dua matriks dapat dijumlahkan jika ukurannya sama. Jika diberikan dua matriks berukuran sama A dan B maka A+B adalah matriks yang diperoleh dengan menjumlahkan entri A dengan entri B yang bersesuaian. Demikian juga untuk pengurangan dua matriks, diperoleh dengan mengurankan entri yang bersesuaian. Cobalah lakukan Penjumlahan A dengan B. [Tampilkan: contoh dua matriks, jumlahkan (perlihatkan proses) Definisi jumlahan secara umum, sengan A, B dua matriks maka (A+B)ij = (A) ij + (B)ij = aij + bij Demikian juga untuk pengurangan A- B Sediakan kotak sbg posisi entri (A + B)ij, kemudian isi dengan jumlahan aij dan bij Sediakan dua matriks dan templae hasil jumlahan untuk diisi mahasiswa] Quiz: berikan dua matriks berukuran sama, jumlahkan berikan dua matriks berukuran beda, jumlahkan aa hasilnya? A + B adalah matriks nol, apakesimpulannya? What two conditions so that the matrix can summand? Answer: ordo two are the same matrix A = [aij] and B = [bij] the same size, A + B defined: (A + B)ij = (A)ij + (B)ij = aij + bij Hal.: 91 Isi dengan Judul Halaman Terkait

example: A = B = A + B = ? Hal.: 92 Isi dengan Judul Halaman Terkait

answer : A + B = + = = Hal.: 93 Isi dengan Judul Halaman Terkait

example: A = B = Find the value A + B! WHY ????? Hal.: 94 Isi dengan Judul Halaman Terkait

Condition 2 matrix can summand : * Have the same ordo Hal.: 95 Isi dengan Judul Halaman Terkait

Exercise 1 Find the value p, q, r, s,and t if given matrix A = matrix B with: A = B = p, q , r, s, t whole number Hal.: 96 Isi dengan Judul Halaman Terkait

Exercise 2 2. B = A = Find the value A+B! Are the results the same? Find the value B+A! Hal.: 97 Isi dengan Judul Halaman Terkait

Solution for number 2 A + B = + = Hal.: 98 Isi dengan Judul Halaman Terkait

Solution for number 2 B + A = + = Hal.: 99 Isi dengan Judul Halaman Terkait

exercise: summation two matrix (next)
K = L = D = C = C + D = ? ? ? D + C = Diberikan matriks K, L, C, D. K dapat dijumlahkan dengan L. Demikian juga C dengan D. Hitunglah C + D, dan K + L. Hitunglah juga D + C dan L + K. Apakah jumlahan dua matriks bersifat komutatif? K + L = ? ? ? L + K = What conclusions you? summation whether the matrix is commutative? Hal.: 100 Isi dengan Judul Halaman Terkait

Quiz: summation two matrix
C + D =… C + E = … A + B = … -8 0 4 7 2 -1 8 4 C = D = 7 2 5 2 6 -1 8 4 E = 7 2 5 2 6 0 0 0 A = 0 0 0 B = Quiz maker 6 -1 2 9 9 8 C +D = Feedback: Hal.: 101 Isi dengan Judul Halaman Terkait

Results scalar time with the matrix
example: 5A = 5x5 5x x1 5x7 5x2 5x3 = A = What is the relationship between H with A? H = H = 50A Perkalian matriks dengan skalar Perkalian matirks A dengan bilangan (skalar c) menghasilkan matriks yang setiap entrinya merupakan hasil kali entri A dengan c. Pada himpunan Mmxn, perkalian matriks dengan skalar bersifat tertutup. [Berikan contoh matriks dan bilangan. Kemudian kalikan. Berikan matriks, sajikan sebagai hasil kali skalar dan matriks lain.Definisi perkalian matriks dengan skalar secara umum. Catatan: Pada himpunan Mmxn, perkalian matriks dengan skalar bersifat tertutup. ] animasikan Quiz: Diberikan suatu matriks A, cA adalah matriks nol. Apa kesimpulan Anda tentang A dan c?Kunci: kasus 1: c=0 dan A matriks sembarang. Kasus 2: A matriks nol dan c bisa berapa saja.  atau sebaiknya dalam ppt? given matrix A = [aij] and scalar c, multiplication scalar cA have entries as follows : (cA)ij = c.(A)ij = caij Note: On the set of Mmxn, matrix multiplication with skalar are closed (the matrix with the same ordo) Hal.: 102 Isi dengan Judul Halaman Terkait

Results scalar time with the matrix (next)
K 3 x 3 K = 4K = 5K = Kalikan K dengan 4 kemudian kalikan dengan 5. Kalikan K dengan suatu skalar sedemikian hingga hasilnya matriks nol. Hal.: 103 Isi dengan Judul Halaman Terkait

Exercise: Results skalar times with the matrix
given that cA is the matrix zero. What is your conclusion about the A and c? 0 0 0 A = example: c = 7 c = 0 A = 7 2 5 2 6 cA = 0*2 0*7 0*2 0*5 0*2 0*6 Misalkan cA adalah matriks nol. Apa kesimpulan Anda tentang A dan c? Terdapat dua kemungkinan: Pertama, c = 0 dan A bisa sembarang matriks, Kedua, A matriks nl, dan c bisa sembarang skalar cA = 7*0 7*0 7*0 0 0 0 = conclusion case 1: c = 0 and A random matrix. case 2: A zero matrix and c how can it. Hal.: 104 Isi dengan Judul Halaman Terkait

Multiplication matrix
B = A = A B = = Perkalian matriks Jika A matriks berukuran kxl, B berukuran lxm, maka hasil kali A dan B adalah matriks C berukuran kxm. Entri baris ke i kolom ke j matriks C adalah jumlahan dari hasil kali entri pada baris i dengan entri di kolom j yang bersesuaian. Sebagai contoh, entri baris pertama kolom pertama AB adalah jumlahan hasil kali baris pertama A Dengan kolom pertama B. Entri baris ke dua kolom pertama AB (yaitu -49) diperoleh dengan menjumlahkan hasil kali baris ke dua A dengan kolom pertama B. Apa syaratnya agar dua matriks dapat dikalikan? Jawabnya adalah: jumlah kolom matrik pertama harus sama dengan jumlah baris matriks kedua. Apakah perkalian matriks tidak bersifat komutatif? Jika hasil kali dua matriks adalah matriks nol, apakah salah satu matriks yang dikalikan adalah matriks nol? t Perpangkatan matriks Perpangkatan matiks merupakan perkalian berulang. Tentu saja perpangkatan hanya dapat dilakukan pada matriks persegi. Pangkat nol dari matriks didefinisikan sebagai matriks identitas. [Definisi dengan bentuk umum  demo perkalian matriks. Highlight di tempat yang tepat] [Berikan contoh matrik 3x4 dan 4x2, kalikan, highlight baris dan kolom yang sedang dkerjakan. Berikan kotak2 untuk menampung hasil kali baris dan kolom. (Anton 28) Berikan contoh matriks yang tidak dapat dikalikan. Diagram ukuran perkalian matriks: A B AB kxl lxm kxm (lengkapi dengan garis] Sifat-sifat perkalian matriks Quiz: A x B apakah sama dengan B x A? Feedback dgn counter example matriks 2x2  atau ppt? AB = O matriks nol, apakah salah satu dari A atau B pasti matriks nol? Feedback: dengan counter example. berikan beberapa pasang matriks, mhs diminta untuk menentukan hasil kalinya jika dapat dikalikan.WS 1: diberikan tabel mahasiswa tick pada cell jika benar Perpngkatanmatriks: A0 matriks identitas. An =…, A n Am =…. Hal.: 105 Isi dengan Judul Halaman Terkait

Multiplication matrix (next)
Definition: If A = [aij] sized m x r , and B = [bij] sized r x n, then matrix result multiply A and B, is C = AB have elemen-elements which is defined as follows : r ∑ aikbkj = ai1b1j +ai2b2j+………airbrj k = 1 (C)ij = (AB)ij = A B AB condition: m x r r x n m x n Misalkan C = AB. Elemen baris ke-i kolom ke j dari C sama dengan jumlahan baris ke i matriks A dengan kolom ke j matriks B. Jika diberikan matriks A dan B, AB terdefinisi jika jumlah kolom A sama dengan jumlah baris B. Perhatikan diagram. Hasil kalinya, yaitu AB mempunyai jumlah baris = jumlah baris A, dan jumlah kolom AB = jumlah kolom B. Diberikan dua matriks, dapatkah kamu menentukan AB dan BA? B = A = Find the value AB and BA Hal.: 106 Isi dengan Judul Halaman Terkait

B = A = A B = = AB terdefinisi karena jumlah kolom A = jumlah baris B. BA tidak terdefinisi, sebab jumlah kolom B tidak sama Dengan jumlah baris A. BA not defined Hal.: 107 Isi dengan Judul Halaman Terkait

Given A and B, defined AB and BA. What your conclusion? A B n x k m x n B A n x k m x n m = k AB and BA square matrix ABmxm ABnxn 2. AB = O zero matrix, if one of A or B certainly zero matrix? Quiz Feedback: A dan B matriks persegi dengan ordo sama Lihat slide Belum tentu, berikan contoh A = B = AB = AB zero matrix, not A or B matrix is zero Hal.: 108 Isi dengan Judul Halaman Terkait

Exercise: Multiplication matrix (next)
Find the value if the time defined AB = ?? AC = ?? BD = ?? CD = ?? DB = ?? B = A = C = D = [Matriksnya muncul dulu baru soalnya] Kerjakan latihan ini, jika masih belum yakin dengan hasilnya beberapa hal dapat kamu lakukan: Membaca buku text, Diskusi dengan teman, Kirim pertanyaan di forum Hal.: 109 Isi dengan Judul Halaman Terkait

The root of matrix example: A = A2 = A3 = A x A2 = Pepangkatan matriks didefinisikan sebagai perkalian berulang. Tentukan A2, kemudian kalikan hasilnya dengan A untuk memperoleh A3. Perpangkatan didefinisikan utnuk matriks-matriks persegi saja. A0 = I An = n factor An+m = An Am A A A …A Hal.: 110 Isi dengan Judul Halaman Terkait

Presented with the SPL of the matrix
SPL in the form of: can be presented in the form of a matrix equation : a11x1 + a12x2 + a13x3 +….. ..a1nxn = b1 a21x1 + a22x2 + a23x3 +…….a2nxn = b2 : am1x1 + am2x2 + am3x3 + ……amnxn = bm a11 a12……...a1n a21 a22 ……..a2n : : : am1 am2…… amn x1 x2 : xn b1 b2 : bn Perkalian matriks dan SPL SpL ini dapat dinyatakan dalam persamaan matriks. Jika A adalah matriks koefisien A, unknown disajikan dalam matriks kolom X, dan konstanta disajikan dalam vektor kolom b. Maka SPL dapat disajikan dalam persamaan Ax = b. A disebut matriks koefisien SPL. = A: matrix coefficients x b Ax = b Hal.: 111 Isi dengan Judul Halaman Terkait

example: Presented with the SPL of the matrix
x1 + 2x2 + x3 = 6 -x2 + x = 1 4x1 + 2x2 + x3 = 4 SPL 1.x1 +2.x x3 0.x x2 + 1.x3 4.x1 +2.x x3 6 1 4 x1 x2 x3 6 1 4 = = = Diberikan SPL, perhatikan langkah-langkah untuk menyajikan SPL tersebut dalam bentuk matriks. Buatlah contoh SPL dan sajikan dalam persamaan matriks. Jika kamu diberi matriks koefisien, dapakah kamu membentuk SPLnya? Memahami konsep ini amat penting untuk memahami konsep terkait di modul-modul yang akan datang. [tunjukkan matriks koefisiennya, perlihatkan proses supaya jelas. Misalnya dengan langkah2 sbb: diberikan persamaan, persamaan dikurung dalam dua matriks, yang sebelah kiri diuraikan (spt pada bgn kanan tanda panah), baru dipecah (seperti pada sebelah kiri panah)]  proses spt pada slide 21 Hal.: 112 Isi dengan Judul Halaman Terkait

multiplication matrix and identity matrix
A.I = = Masih ingat hasil kali matriks dengan matriks identitas? Marilah kita kalikan A dengan matriks identitas I, baik dari kiri maupun dari kanan. Bagaimana hasilnya? Apa kesimpulanmu? Perkalian matriks persegi dengan matriks identitas berukuran sama bersifat komutatif. Hasil kalinya sama dengan matriks A. I.A = = Hal.: 113 Isi dengan Judul Halaman Terkait

multiplication matrix and identity matrix AB = A and BA = A, what your conclusion? = = QUIZMAKER Feedback: A dan B matriks persegi dengan ordo sama B adalah matriks identitas A I I A A = = AB = A and BA = A, then B = I (I identity matrix) Hal.: 114 Isi dengan Judul Halaman Terkait

Inverse matriks B is inverse from matriks A, if AB = BA = I identity matrix, wrote B = A-1 A A-1 A-1 A I = = Example ½ -½ -½ 1 ½ -½ -½ 1 = = A A-1 A-1 A I Diberikan matriks persegi A, matriks manakah jika dikalikan dengan A hasilnya matriks identitas? Jawabnya adalah inverse dari A. Marilah kita definisikan inverse matriks. B adalah inverse dari matriks A, jika AB = BA = I matriks identitas, ditulis B = A inverse (seperti pangkat -1). Perhatikan contoh berikut: A kali A inverse sama dengan A inverse kali A sama dengan matriks identitias I. B kali B inverse sama dengan B inverse kali B sama dengan matriks identitas I. Masalah berikutnya adalah: Apakah setiap matriks mempunyai inverse? Jika mempunyai, bagaimana menentukannya? Apakah inverse matriks (jika ada0 adalah tungga? ½ -½ -½ -½ ½ -½ -½ -½ = = B B-1 B-1 B I Hal.: 115 Isi dengan Judul Halaman Terkait

Inverse matriks 2x2 ½ -½ -½ 1 = A A-1 I a b c d A-1 = Marilah kita menentukan inverse matriks 2x2. Perhatikan bahwa A kali A inverse adalah I. Jika A diketahui, bagaimanan menentukan A inverse? A-1 d -b -c a 1 ad - bc = = If ad –bc = 0 then A haven’t inverse. Hal.: 116 Isi dengan Judul Halaman Terkait

example: Inverse matrix 2x2
A = Berikut contoh penerapan rumus menentukan inverse matriks 2x2 A-1 = = I Hal.: 117 Isi dengan Judul Halaman Terkait

Quiz: inverse matriks 1. When matrix does not have any inverse? ad-bc = 0 2. Find the value inverse matriks in the below a. 2/ /5 -1/ /3 a. b. b. haven’t inverse QUIZMAKER c. c. haven’t inverse d. d. Hal.: 118 Isi dengan Judul Halaman Terkait

Quiz: Please fill in the dots below A symmetry then A + AT= …….. ((AT)T)T = ……. (ABC)T = ……. ((k+a)A)T = …..... (A + B + C)T = ………. QUIZMAKER Key: 2A AT CTBTAT (k+a)AT AT + BT + CT Hal.: 119 Isi dengan Judul Halaman Terkait

Properties Inverse Matrix
Inverse from matrix if there is one: Jika B = A-1 dan C = A-1, maka B = C (A-1)-1 2. (A-1)-1 = A A = ½ -½ -½ 1 ? = Sifat-sifat matriks inverse Jika A mempunyai inverse, maka inversenya tunggal. Inverse dari inverse matriks A adalah A sendiri. Matriks inverse dari matriks pangkat n sama dengan inverse matriks dipangkatkan n. 1-5 dalam bentuk rumus, kemudian contoh-contoh sederhana Quiz: menentukan inverse matriks 2x2, inverse dari transposenya, berikan 2 matriks, satu rtogonal satu tidak, identifikasi. Untuk matriks ort, hitung determinan. Link ke bukti sifat 1, 2 ½ -½ -½ 1 A-1 = A-1 A Hal.: 120 Isi dengan Judul Halaman Terkait

Properties (next) If A have inverse then An have inverse and (An)-1 = (A-1)n, n = 0, 1, 2, 3,… A = ½ -½ -½ 1 A-1 = A3 = = (A3)-1 = Jika A adalah matriks persegi yang mempunyai inverse, maka menghitung inverse kemudian memangatkan hasilnya sama dengan memangkatkan dahulu kemudian dihitung inversenya. same (A-1)3 = ½ -½ -½ 1 ½ -½ -½ 1 ½ -½ -½ 1 = Hal.: 121 Isi dengan Judul Halaman Terkait

Properties (next) If k scalar not zero, then (kA)-1 = 1/k A-1 (5A) = 5 = (5 A)-1 = same Menghitung inverse dari kA dapat dilakukan dengan menghitung inverse dari A kemudan dikalikan dengan 1/k. ½ -½ -½ 1 = 1/5 (A)-1 = 1/5 Hal.: 122 Isi dengan Judul Halaman Terkait

Properties (next) (AB)-1 = B-1 A-1 B = B-1 = ½ /4 ½ ¾ A = (AB)-1 = -1 = Jika A dan B dapat dikalikan dan masing-masing mempunyai inverse, maka AB juga mempunyai inverse. Inverse AB sama dengan hasil kali B inverse deang A inverse (urutan terbalik) ½ /4 ½ ¾ ½ -½ -½ 1 B-1 A-1 = = ½ -½ -½ 1 ½ /4 ½ ¾ A-1 B-1 = = Hal.: 123 Isi dengan Judul Halaman Terkait

Complete equation Linear System With Two Variables and Determinan Matrix Method

General form from linear equation system with two variables
Hal.: 125 Isi dengan Judul Halaman Terkait

There is method in solving linear equation system with two variables?
Method of solution of SPL There is method in solving linear equation system with two variables? Hal.: 126 Isi dengan Judul Halaman Terkait

Method of solution of SPL Methods in solving linear equation system with two variables : Graphic Method Substitution Method Elimination Method Combined methods (the Elimination and Substitution) Determinan Matrix Method Hal.: 127 Isi dengan Judul Halaman Terkait

Linear equation system
Rules Cramer Linear equation system With the elimination method can be obtained from the value of X: Hal.: 128 Isi dengan Judul Halaman Terkait

Rules Cramer Linear equation system With the elimination method can be obtained from the value of Y: Hal.: 129 Isi dengan Judul Halaman Terkait

Method Matrix determinant Linear equation system Determinant D from coefficient matrix of linear equation system is : Hal.: 130 Isi dengan Judul Halaman Terkait

Method Matrix determinant Explore the value X that meet the equality : Hal.: 131 Isi dengan Judul Halaman Terkait

Method Matrix determinant Explore the value Y that meet the equality : Hal.: 132 Isi dengan Judul Halaman Terkait

The value from X and Y exist, if: D≠0 or or or Hal.: 133 Isi dengan Judul Halaman Terkait

example Define the set of the settlement system of equality: Hal.: 134 Isi dengan Judul Halaman Terkait

Solution from linear equation system
Determinant D Hal.: 135 Isi dengan Judul Halaman Terkait

The value X that meet the : Hal.: 136 Isi dengan Judul Halaman Terkait

The value Y that meet the : Hal.: 137 Isi dengan Judul Halaman Terkait

So the collective system of settlement is
HP So the collective system of settlement is {(3,-4)} Hal.: 138 Isi dengan Judul Halaman Terkait

Exercise Define the set of the settlement system of the following equation: Hal.: 139 Isi dengan Judul Halaman Terkait

Homework Hal.: 140 Isi dengan Judul Halaman Terkait

Oleh : SARTIM S, S.Pd. GURU MATEMATIKA SMKN 30 JAKARTA
EQUATION AND INEQUALITY Oleh : SARTIM S, S.Pd. GURU MATEMATIKA SMKN 30 JAKARTA

Competency standards :
Solve problems related to the system of linear and inequality and quadratic. Hal.: 142 Isi dengan Judul Halaman Terkait

Learning materials : Equation and inequality linear. Quadratic equation and inequality. Roots of the quadratic equation and properties. Establishing new quardtic equality. Implementation of quadratic equality and inequality in problem solving Skills Program . Hal.: 143 Isi dengan Judul Halaman Terkait

Basic kompetence 1 : Determine the collective settlement of linear equality and inequality. Hal.: 144 Isi dengan Judul Halaman Terkait

Learning objectives Explain the linear equation and inequality. Complete equality and inequality one linear variable. Complete linear equation system of two variables. Problem solving skills program using the linear equation and inequality . Hal.: 145 Isi dengan Judul Halaman Terkait

Linear equation example
Finished equation below : 1. 3x -5 = 4 2. 3(3x + 6) = 5x + 2 3. ½ x + 2/3 = 3/4 Hal.: 146 Isi dengan Judul Halaman Terkait

Answer no 1 and 2 1. 3x – 5 = 4 3x = 4 + 5 3x = 9 x = 3 2. 3(3x + 6) = 5x + 2 9x = 5x + 2 9x – 5x = 2 – 18 4x = -16 x = -4 Hal.: 147 Isi dengan Judul Halaman Terkait

Answer no 3 ½ x + 2/3 = ¾ x 12 6 x = 9 6x = 9 – 8 6x = 1 x = 1/6 Hal.: 148 Isi dengan Judul Halaman Terkait

Linear inequality example
Finished inequality below : 1. 3x -5 > 4 2. 3(3x + 6) < 5x + 2 3. ½ x + 2/3 ≥ 3/4 Hal.: 149 Isi dengan Judul Halaman Terkait

Answer no 1 and 2 1. 3x – 5 > 4 3x > 4 + 5 3x > 9 x > 3 2. 3(3x + 6) < 5x + 2 9x < 5x + 2 9x – 5x < 2 – 18 4x < -16 x < -4 Hal.: 150 Isi dengan Judul Halaman Terkait

Answer no 3 ½ x + 2/3 ≥ ¾ x 12 6 x ≥ 9 6x ≥ 9 – 8 6x ≥ 1 x ≥ 1/6 Hal.: 151 Isi dengan Judul Halaman Terkait

Two varibels SPL example
Find the value x and y satisfy the following linear equation system with a mixture of methods (elimination and substitution): 2x + 3y = 18 5x - y = 11 Hal.: 152 Isi dengan Judul Halaman Terkait

Answer : 2x + 3y = 18 |x 1| 2x + 3y = 18 5x - y = 11 |x 3|15x – 3y = 33 + 17x = 51 x = 3 2x + 3y = 18 y = 18 6 + 3y = 18 3y = 18 – 6 3y = 12 y = 4 Hal.: 153 Isi dengan Judul Halaman Terkait

Exercise 1.finished equation and inequality below : a. 2x – 6 = 8 b. 3x + 4 = x – 8 c. 5(2x + 3) = 7x – 3 d. ½ x – 2/3 = 5/6 e. 5x + 6 > 11 f. 6x – 8 ≤ 2x + 4 g. 2( 3x – 4) ≥ 9x + 7 Hal.: 154 Isi dengan Judul Halaman Terkait

Exercise (next) 2. find the value x and y from equation system below : a. 3x + 4y = 13 dan 5x – 3y = 12. b. 4x – 2y = 0 dan 6x + 7y = 20. Hal.: 155 Isi dengan Judul Halaman Terkait

Quadratic equation and inequality

:: part of this material :: 2. Find the root quadratic equation 1. Quadratic equation definition 3. Kind of quadratic equation root 4. Formula sum & multiply quadratic equation root 5. Quadratic inequality Hal.: 157 Isi dengan Judul Halaman Terkait

Basic kompetence 2 : Determine the settlement of the collective quadratic equation and inequality. Hal.: 158 Isi dengan Judul Halaman Terkait

Learning objectives Explain the quadratic equation and inequality. Explain the root and properties quadratic equation. Complete quadratic equality and inequality. Hal.: 159 Isi dengan Judul Halaman Terkait

1. quadratic equation definition
`an equation where the highest rank of variables, namely two` General form of quadratic equation : with Hal.: 160 Isi dengan Judul Halaman Terkait

Quadratic equation example a = 2, b = 4, c = -1 a = 1, b = 3, c = 0 a = 1, b = 0, c = -9 Determine the settlement of quadratic equation in x means that the search for value so that if the value of x in the equation is disubsitusikan, then the equation will be valued properly. Settlement of square root is also called square-root of the equation. Hal.: 161 Isi dengan Judul Halaman Terkait

2. determine the root of the quadratic equation
There are three ways to determine the square root of the equation, namely : Factorization Complete perfect square Quadratic formula (formula a b c) Hal.: 162 Isi dengan Judul Halaman Terkait

Factorization To complete the equation ax² + bx + c = 0 with factorization, First time find two numbers that meet the following requirements as . Time results is same ac sum is same b Suppose the two numbers that are eligible and , and then Basic principles that are used to complete the quadratic equation Factorization is the nature of multiplication, namely : If ab = 0, then a = 0 or b = 0 . So, will change if the formula or standard form of equation square ax² + bx + c = 0 . for a = 1 factorization form ax² + bx + c = 0 to be : for a ≠ 1 Lanjut Hal.: 163 Isi dengan Judul Halaman Terkait

example : Complete the following quadratic equation with factorization : 1. x2 + 7x + 12 =0 2. x2 - 4x – 21 = 0 3. 6x2 -7x – 20 = 0 Hal.: 164 Isi dengan Judul Halaman Terkait

answer no 1 and 2 . 1. X2 + 7x + 12 = 0 (x + 3)(x + 4) = 0 X1 = -3, x2 = -4 2. x2 – 4x – 21 = 0 (x + 3)(x – 7) = 0 x1 = -3, x2 = 7 Hal.: 165 Isi dengan Judul Halaman Terkait

answer no 3 . 3. 6x2 – 7x -20 = 0 (6x – 15)(6x + 8) = 0 6 (2x – 5)(3x + 4) = 0 x1 = 5, x2 = -4 Hal.: 166 Isi dengan Judul Halaman Terkait

complete perfect square Quadratic equation ax² + bx + c = 0, change to be general form perfect quadratic with below : Ensure the coefficient of x² is 1, if not value 1 divide the number of such coefficients to the value 1. Add segment with the left and right half Coefficient of x and make to square . Make a left segment of perfect square, While the right segment simplified . Lanjut Hal.: 167 Isi dengan Judul Halaman Terkait

Example : Complete the following quadratic equation with complete square (perfect square) : 2x2 - 8 x - 42 = 0 Hal.: 168 Isi dengan Judul Halaman Terkait

answer : 2x2 -8x -42 = 0 : 2 X2 – 4x – 21 = 0 X2 – 4x = X2 – 4x = 21 X2 – 4x + 22 = ( x – 2 ) = 25 X – = +/- 5 X1 = = 7 X2 = = -3 Hal.: 169 Isi dengan Judul Halaman Terkait

quadratic formula (formula a b c) By using the rules complete perfect square which has been published previously, you can search in the formula for complete quadratic equation . if and namely root quadratic equation ax² + bx + c = 0, then : Hal.: 170 Isi dengan Judul Halaman Terkait

Exercise Complete the following quadratic equation with factorization, perfect square and with formulas : 1. x2 + 5x + 4 = 0 2. x2 + 6x = 40 3. 6x2 – 11x – 10 = 0 Hal.: 171 Isi dengan Judul Halaman Terkait

3. types of quadratic equation Isi dengan Judul Halaman Terkait
value from b² - 4ac namely diskriminant, that is D = b² - 4ac . Some kind of square root of the equation based on the value of D . If D > 0, then quadratic equation has two real roots Different. If D = 0, then quadratic equation has two real roots the same or have the root is often called the twin (same) . c. If D < 0, then square root of the equation that does not have any Real (imaginary) . Hal.: 172 Isi dengan Judul Halaman Terkait

4. Formula of the sum and the product of roots of quadratic equations
If the note is the root of the second, then obtained: If both roots are multiplied, then obtained : Both of the above mentioned formula of the sum and the product of roots of quadratic equations. Hal.: 173 Isi dengan Judul Halaman Terkait

1. The sum of roots of quadratic equations: x1 + x2 = + = = = Hal.: 174 Isi dengan Judul Halaman Terkait

2. The product of roots of quadratic equations x1 . x2 = . = = = = = Hal.: 175 Isi dengan Judul Halaman Terkait

The sum and the product of roots of root quadratic equations If x1 and x2 of the both are roots of quadratic equations ax2 + bx +c = 0 (a  0), the sum and the product of roots of quadratic equations determined with the formula x1 + x2 = x1 . x2 = Formula the sum and the product of roots can using for distinguish the characteristics of the root-root of the quadratic equation has two different real roots. Such as the following: Hal.: 176 Isi dengan Judul Halaman Terkait

The sum and the product of roots of quadratic equations: 1. One is the root of the root of the opponent or the other are often perceived to be radical opposite : x1 = - x2  x1 + x2 = 0  = 0  b = 0 2. One is the root of the inverse root of the other or are often perceived to be radical the reverse : x1 =  x1 . x2 = 1  = 1  a = c Hal.: 177 Isi dengan Judul Halaman Terkait

The sum and the product of roots of quadratic equations: 3. One root same with 0: x1 = 0  x1 . x2 =  x1 + x2 =  x1 + x2 =  (0) + x2 =  x2 = (0) x2 =  =  c = 0 Hal.: 178 Isi dengan Judul Halaman Terkait

The sum and the product of roots of quadratic equations: 4. The second root has the same sign or are often perceived to be radical with the same : x1  0 and x2  0 or x1  0 and x2  0 x1 . x2  0  > 0, a and c marked the same 5. Both root does not have any signs that are often perceived to be the same or roots of the different signs : x1  0 and x2 0 or x1  0 and x2  0 x1 . x2  0   0, a and c marked the same Hal.: 179 Isi dengan Judul Halaman Terkait

5. quadratic inequality quadratic inequality is inequality that have a variable with the highest rank of the two . Steps to seek settlement of the collective quadratic inequality : Indicate quadratic inequality in the form of quadratic equation (made with the right segment 0) . Find the square root of the equation. Create a line number which is the roots, set the sign (positive or negative) on each interval . Set of intervals obtained from the settlement that meets these inequality. Hal.: 180 Isi dengan Judul Halaman Terkait

Exercise Finished : 1. x2 + 5x + 4 > 0 2. x2 + 6x ≤ 40 3. 6x2 – 11x – 10 ≥ 0 Hal.: 181 Isi dengan Judul Halaman Terkait

Basic competence 3 : Applying quadratic equation and inequality Hal.: 182 Isi dengan Judul Halaman Terkait

Learning objectives Establishing quadratic equality based on roots is known. Establishing quadratic equality based on roots of the quadratic equation others. Problem solving skills program based on the quadratic equation and inequality. Hal.: 183 Isi dengan Judul Halaman Terkait

Apersepsi: general form quadratic equation, namely: ax2 + bx +c = 0 to determine formula of the sum and the product of roots of quadratic equations, we remember that roots from equation ax2 + bx +c = 0 (a  0) can determine with formula x1,2 = x1 = x2 = Hal.: 184 Isi dengan Judul Halaman Terkait

Find the roots of : x2 - 5x + 6 = 0
Problem 1 Find the roots of : x2 - 5x + 6 = 0 Horeeee I can………..! Hal.: 185 Isi dengan Judul Halaman Terkait

Find the quadratic equation, if roots is x1 = 2 and x2 = 3
Problem 2 Find the quadratic equation, if roots is x1 = 2 and x2 = 3 ????? Emmm what equation??? Hal.: 186 Isi dengan Judul Halaman Terkait

Problem solving Find the roots of x2-5x+6=0 Answer : (x-2)(x-3)=0 x-2=0 or x-3=0 x= x=3 Hp {2,3} Find the quadratic equation, if roots is x1=2 & x2=3 then: x= x=3 x-2=0 or x-3=0 (x-2)(x-3)=0 x2-2x-3x+6=0 x2-5x+6=0 Hal.: 187 Isi dengan Judul Halaman Terkait

Note Find the roots of x2-3x+2=0 answer: (x-1)(x-2)=0 x-1=0 or x-2=0 x= x=2 Hp {1,2} Find the quadratic equation, if roots is x1=1 & x2=2 then: x= x=2 x-1=0 or x-2=0 (x-1)(x-2)=0 x2-2x-x+2=0 x2-3x+2=0 Hal.: 188 Isi dengan Judul Halaman Terkait

Note Find the quadratic equation, if roots is x1=-1 & x2=3 then: x= x=3 x-(-1)=0 or x-3=0 (x+1)(x-3)=0 x2+x-3x-3=0 x2-2x-3=0 Find the roots of x2-2x-3=0 Answer : x2-2x-3=0 (x-3)(x+1)=0 x-3=0 or x+1=0 x= x=-1 Hp {-1,3} Hal.: 189 Isi dengan Judul Halaman Terkait

What you get…? If quadratic equation have roots of x1=a dan x2=b then quadratic equation is : x=a x=b x-a=0 or x-b=0 (x-a)(x-b)=0 x2-ax-bx+ab=0 x2-(a+b)x+ab=0 Hal.: 190 Isi dengan Judul Halaman Terkait

Conclusion To arrange a quadratic equation has been known radical with x1=a or x2=b then equality is: (x-a)(x-b)=0 or x2-(a+b)x+a.b=0 Hal.: 191 Isi dengan Judul Halaman Terkait

Note If x1 and x2 is roots of quadratic equation ax2 + bx +c = 0 (a  0), the sum and the product of roots of quadratic equations is : x1 + x2 = x1 . x2 = Hal.: 192 Isi dengan Judul Halaman Terkait

If x1 and x2 roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0).
The opposite of roots (x1 = - x2 )  b = 0. The reversal of roots ( x1 = )  a = c. A root equal to zero ( x1 = 0)  c = 0 and x2 = A both of roots the same sign  > 0. A both of roots the different sign  < 0. Hal.: 193 Isi dengan Judul Halaman Terkait

Note Complete quadratic equations Quadratic equation ax2 + bx + c = 0 Roots x1, x2 Prepare quadratic equations Hal.: 194 Isi dengan Judul Halaman Terkait

Composing quadratic equations if the roots are given
Product of factor (x- x1) (x – x2) = 0 b. Using the sum and the product of roots x2 – (x1 + x2) x + (x1 . x2) = 0 Hal.: 195 Isi dengan Judul Halaman Terkait

Example 1: Roots of quadratic equations 3x2 + 6x – 8 = 0 are p and q. Compose of quadratic equations if roots are and . Answer : remember!! Formula the sum and the product of roots: x2 – (x1 + x2)x + x1 . x2 = 0 Given quadratic equation 3x2 + 6x – 8 = 0 Of roots of p and q, then: Hal.: 196 Isi dengan Judul Halaman Terkait

Advanced answer instance: Quadratic equations have requested of roots of x1 and x2, then Hal.: 197 Isi dengan Judul Halaman Terkait

Advanced answer Substitute and to equation x2 – (x1 + x2)x + x1 . x2 = 0 to get: So, quadratic equation is 8x2 – 6x – 3 = 0 Hal.: 198 Isi dengan Judul Halaman Terkait

example 2: The qudratic equation 3x2 + x – 2 = 0, prepare quadratic equation if roots is a square from roots of equation to determine! Answer : remember!! Formula the sum and the product of roots: x2 – (x1 + x2)x + x1 . x2 = 0 Given quadratic equation 3x2 + x – 2 = 0, then: Hal.: 199 Isi dengan Judul Halaman Terkait

Advanced answer instance: asked the equality y2 – (y1 + y2)y + y1 . y2 = 0 with roots y1 and y2 then, Hal.: 200 Isi dengan Judul Halaman Terkait

Advanced answer Substitution: and to equation y2 – (y1 + y2)y + y1 . y2 = 0 to get: So, quadratic equation is 9y2 – 13y + 4 = 0 Hal.: 201 Isi dengan Judul Halaman Terkait

Example 3 Given α and β is the roots of equation 4x2 – 3x – 2 = 0. find the quadratic equation if the roots of (α + 3) and (β + 3)! answer: Given quadratic equation 4x2 – 3x – 2 = 0 of the roots of α and β, then: α + β = = and α . β = = Instance quadratic equation namely, x2 – (x1 + x2)x + x1.x2 = 0 have the roots of x1 and x2, then x1 = (α + 3) and x2 = (β + 3), then we get: x1 + x2 = (α + 3) + (β + 3) = (α + β) + 6 = = Hal.: 202 Isi dengan Judul Halaman Terkait

Advanced answer x1 . x2 = (α + 3) . (β + 3) = (α . β) + 3(α + β) + 9 = ( ) + 9 = = Substitution x1 + x2 = and to equation x2 – (x1 + x2)x + x1.x2 = 0, and get x2 – x = 0, both segment to multiply 4 4x2 – 27x + 43 = 0 So, equality that is required 4x2 – 27x + 43 = 0. x1 . x2 = Hal.: 203 Isi dengan Judul Halaman Terkait

Exercise Set equality x2 + bx + c = 0, a ≠ 0. composing new quadratic equation the roots of: a. opposite with the roots of given b. reversal with the roots of given c. larger n times from the roots of given d. m more than from the roots of given e. constitutes exponent 3 from the roots of given. Set equality x2 – 3x + 1 = 0. Composing new quadratic equation the roots of: c. larger 3 times from the roots of given d. 3 more than from the roots of given Hal.: 204 Isi dengan Judul Halaman Terkait

Implementation Square Equations in daily life Hal.: 205 Isi dengan Judul Halaman Terkait

Ways to solve quadratic equations are :
Factoring Completing the square Formula Hal.: 206 Isi dengan Judul Halaman Terkait

Story problem related to the quadratic equation
Budi will answer the question from their teacher about election with both the number the following : difference both the number is two and the product both the number is 168. how the number is ? Hal.: 207 Isi dengan Judul Halaman Terkait

Matter how complete the story
Reading problems Translate in mathematical model Finishing Check to result Hal.: 208 Isi dengan Judul Halaman Terkait

Finishing Budi will answer the question from their teacher about election with both the number the following : Difference both the number is two and the product both the number is 168. how the number is ? Given : ∙ Election both the number ∙ Difference is 2 ∙ The product is 168 Asked : how the number is ? Hal.: 209 Isi dengan Judul Halaman Terkait

next Budi will answer the question from their teacher about election with both the number the following : Difference both the number is two and the product both the number is 168. How the number is ? answer : Instance number is a and b a – b = 2 ………………1) a . b =168 ………………2) from equation 2) a . b =168 => a = 168 / b substitution to equation 1) a-b = 2 (168 / b) – b = 2 b2 + 2b – 168 = 0 (b - 12)(b + 14) = 0 Hal.: 210 Isi dengan Judul Halaman Terkait

Next check ∙ a – b = 2 14 – 12 = 2 ∙ a . b = = 168 For b – 12 =0 b = 12 For b +14 = 0 b = - 14 Remember number always positive then value b to use is 12 a – b = 2  a – 12 = 2  a = 14 So, number is 12 and 14 Hal.: 211 Isi dengan Judul Halaman Terkait

by : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA

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by : SARTIM S, S.Pd. mathematics teacher SMKN 30 JAKARTA

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