Grade 8 Algebraic Identities

Transcription

1 ID : ww-8-algebraic-identities [1] Grade 8 Algebraic Identities For more such worksheets visit Answer t he quest ions (1) There are two numbers such that the sum of their squares is 25 and sum of the numbers is 7. Find their product. (2) Solve the f ollowing using the standard identity (x + a) (x + b) = x 2 + (a + b)x + ab A) B) C) D) (3) Solve the f ollowing using the standard identity (a+b) (a-b) = a 2 - b 2 A) B) C) D) (4) If, f ind the value of p 2 - q 2. (5) There are two numbers such that sum of the numbers is 25 and the dif f erence of their squares is 175. Find their dif f erence. (6) If 3(p 2 + q 2 + r 2 ) = (p + q + r) 2, f ind the value of p + q - 2r. (7) If xy = -8, and -x - 2y = 0, f ind value of x 2 + 4y 2. (8) If = 8, f ind the value of. (9) If (u - 5) 2 + (v - 5) 2 + (w - 3) 2 + (x - 5) 2 + (y - 3) 2 = 0, f ind the value of u + v + w + x + y. (10) Solve the f ollowing using standard identities A) C) 21 2 B) D) Choose correct answer(s) f rom given choice (11) If, f ind the value of a. 2 b. 1/2 c. 0 d. 1.

2 (12) Find the value of ( )2 - ( ) using standard identities. ID : ww-8-algebraic-identities [2] a. 40 b. 2 c. 20 d. 30 (13) Find the value of (35.7)2 - (24.3) using standard identities. a. 60 b. 6 c. 70 d. 600 (14) Solve the f ollowing using the standard identity a 2 - b 2 = (a+b) (a-b) a b c d Fill in the blanks (15) If x 2 + y 2 = 41 and xy = 20, the value of 5(x + y) 2 - (x - y) 2 = Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : ww-8-algebraic-identities [3] (1) 12 Let s assume the two numbers be x and y. It is given that, sum of their squares is 25. Theref ore, x 2 + y 2 = (1) Also the sum of the numbers is 7. Theref ore, x + y = 7 On squaring both sides we get: (x + y) 2 = 49 x 2 + y 2 + 2xy = 49...[Since, (x + y) 2 = x 2 + y 2 + 2xy] xy = 49...[From eqution (1)] 2xy = xy = 24 xy = 24 2 xy = 12 Step 4 Thus, their product is 12. (2) A) We have been asked to f ind the value of using the f ollowing identity: (x + a) (x + b) = x 2 + (a + b)x + ab. Let us think of two simple numbers whose sum is 105. Two such simple numbers are 100 and 5. Similarly, two simple numbers whose sum is 104 are 100 and 4. Thus, = { (5)} { (4)} = {(5) + (4)} (5)(4)...[Using the identity (x + a) (x + b) = x 2 + (a + b)x + ab] = (9)(100) + (20) = (900) + (20) = Theref ore, the result is

4 B) ID : ww-8-algebraic-identities [4] We have been asked to f ind the value of using the f ollowing identity: (x + a) (x + b) = x 2 + (a + b)x + ab. Let us think of two simple numbers whose sum is 996. Two such simple numbers are 1000 and -4. Similarly, two simple numbers whose sum is 995 are 1000 and -5. Thus, = { (-4)} { (-5)} = {(-4) + (-5)} (-4)(-5)...[Using the identity (x + a) (x + b) = x 2 + (a + b)x + ab] = (-9)(1000) + (20) = (-9000) + (20) = Theref ore, the result is C) We have been asked to f ind the value of using the f ollowing identity: (x + a) (x + b) = x 2 + (a + b)x + ab. Let us think of two simple numbers whose sum is Two such simple numbers are 1000 and 4. Similarly, two simple numbers whose sum is 999 are 1000 and -1. Thus, = { (4)} { (-1)} = {(4) + (-1)} (4)(-1)...[Using the identity (x + a) (x + b) = x 2 + (a + b)x + ab] = (3)(1000) + (-4) = (3000) + (-4) = Theref ore, the result is

5 D) ID : ww-8-algebraic-identities [5] We have been asked to f ind the value of using the f ollowing identity: (x + a) (x + b) = x 2 + (a + b)x + ab. Let us think of two simple numbers whose sum is Two such simple numbers are 1000 and 4. Similarly, two simple numbers whose sum is 998 are 1000 and -2. Thus, = { (4)} { (-2)} = {(4) + (-2)} (4)(-2)...[Using the identity (x + a) (x + b) = x 2 + (a + b)x + ab] = (2)(1000) + (-8) = (2000) + (-8) = Theref ore, the result is (3) A) 9919 We have been asked to f ind the value of using the f ollowing identity: (a+b) (a-b) = a 2 - b 2. Let us try to think of two numbers whose sum is 109 and dif f erence is 91. Two such numbers are 100 and 9. Thus, = ( ) (100-9) = [Using the identity (a+b) (a-b) = a 2 - b 2 ] = = 9919 Theref ore, the result is 9919.

6 B) 9936 ID : ww-8-algebraic-identities [6] We have been asked to f ind the value of using the f ollowing identity: (a+b) (a-b) = a 2 - b 2. Let us try to think of two numbers whose sum is 108 and dif f erence is 92. Two such numbers are 100 and 8. Thus, = ( ) (100-8) = [Using the identity (a+b) (a-b) = a 2 - b 2 ] = = 9936 Theref ore, the result is C) 9951 We have been asked to f ind the value of using the f ollowing identity: (a+b) (a-b) = a 2 - b 2. Let us try to think of two numbers whose sum is 107 and dif f erence is 93. Two such numbers are 100 and 7. Thus, = ( ) (100-7) = [Using the identity (a+b) (a-b) = a 2 - b 2 ] = = 9951 Theref ore, the result is 9951.

7 D) ID : ww-8-algebraic-identities [7] We have been asked to f ind the value of using the f ollowing identity: (a+b) (a-b) = a 2 - b 2. Let us try to think of two numbers whose sum is 1009 and dif f erence is 991. Two such numbers are 1000 and 9. Thus, = ( ) (1000-9) = [Using the identity (a+b) (a-b) = a 2 - b 2 ] = = Theref ore, the result is (4) 8 3 (5) 7 Let s assume the two numbers be x and y. It is given that sum of the numbers is 25. Theref ore, x + y = (1) Also the dif f erence of their squares is 175. Theref ore, x 2 - y 2 = (2) Step 4 Now, their dif f erence = x - y = x2 - y 2...[Since, (x - y)(x + y) = x 2 - y 2 ] x + y = 175 = [From equation (1) and (2)] Step 5 Thus, their dif f erence is 7. (6) 0

8 (7) 32 ID : ww-8-algebraic-identities [8] It is given that: xy = (1) It is also given that: -x - 2y = 0 On squaring both sides we get: ( -x - 2y) 2 = 0 (-1x) 2 + (-2y) (-1x) (-2y) = 0...[Since, (a + b) 2 = a 2 + b 2 + 2ab] 1x 2 + 4y 2 + (4)xy = 0 x 2 + 4y 2 + (4)(-8) = 0...[From equation (1)] x 2 + 4y 2 = 32 Thus, the value of x 2 + 4y 2 is 32. (8) 6 If we assume, a = x, and b = 1/x, we can use standard algebraic identities which specif ies relation between a - b and a 2 + b 2 By using identity, (a - b) 2 = a 2 + b 2-2ab ( ) 2 = x 2 + ( 1 x ) 2-2 x 1 x ( ) 2 = 8-2 = 6 Theref ore, the value of is 6.

9 (9) 21 ID : ww-8-algebraic-identities [9] Given (u - 5) 2 + (v - 5) 2 + (w - 3) 2 + (x - 5) 2 + (y - 3) 2 = 0 It means the sum of (u - 5) 2, (v - 5) 2, (w - 3) 2, (x - 5) 2 and (y - 3) 2 is equals to 0. We know that the square of a number cannot be negative. Theref ore, the sum of these non-negative numbers (u - 5) 2, (v - 5) 2, (w - 3) 2, (x - 5) 2 and (y - 3) 2 can be zero only if all of them are also equal to zero. Now, (u - 5) 2 = 0 u - 5 = 0 u = 5 Similarly, v = 5, w = 3, x = 5, y = 3. Step 4 Thus, the value of u + v + w + x + y = = 21 (10) A) 441 Use the standard identities here For example (a+b) 2 = a 2 + b 2 +2ab Similarly,(a-b) 2 = a 2 + b 2-2ab Take the last question here, which is 21 2 Now, 21 = Theref ore, 21 2 = (20 + 1) = (2 x 20 x 1) 21 2 = = 441 B) 7921 Use the standard identities here For example (a+b) 2 = a 2 + b 2 +2ab Similarly,(a-b) 2 = a 2 + b 2-2ab Take the last question here, which is 89 2 Now, 89 = 90-1 Theref ore, 89 2 = (90-1) = (2 x 90 x 1) 89 2 = = 7921

10 C) ID : ww-8-algebraic-identities [10] Use the standard identities here For example (a+b) 2 = a 2 + b 2 +2ab Similarly,(a-b) 2 = a 2 + b 2-2ab Take the last question here, which is Now, 399 = Theref ore, = (400-1) = (2 x 400 x 1) = = D) Use the standard identities here For example (a+b) 2 = a 2 + b 2 +2ab Similarly,(a-b) 2 = a 2 + b 2-2ab Take the last question here, which is Now, 501 = Theref ore, = ( ) = (2 x 500 x 1) = = (11) a. 2

11 (12) c. 20 ID : ww-8-algebraic-identities [11] We have been asked to f ind the value of ( )2 - ( ) 2 identities using standard Now, ( ) 2 - ( ) = ( )( ) [By using the identity a 2 - b 2 = (a + b)(a - b) in the numerator] = = 20 Theref ore, the value of ( )2 - ( ) is 20. (13) a. 60 We have been asked to f ind the value of (35.7)2 - (24.3) using standard identities. Now, (35.7) 2 - (24.3) 2 = 11.4 b) in the numerator] = 11.4 = 60 ( )( ) 11.4 [By using the identity a 2 - b 2 = (a + b)(a - Theref ore, the value of (35.7)2 - (24.3) is 60.

12 (14) b ID : ww-8-algebraic-identities [12] We have been asked to f ind the value of using the f ollowing identity: a 2 - b 2 = (a + b)(a - b). Applying the identity, we can write as: (96 + 4)(96-4) = = 9200 Theref ore, the result is (15) 404 It is given that, x 2 + y 2 = 41 and xy = 20 Now, 5(x + y) 2 - (x - y) 2 = 5(x 2 + y 2 + 2xy) - (x 2 + y 2-2xy) = 5x 2 + 5y xy - x 2 - y 2 + 2xy = 4x 2 + 4y xy = 4(x 2 + y 2 ) + 12xy = 4(41) + 12(20) = 404 Thus, the value of 5(x + y) 2 - (x - y) 2 is 404.