ID : ae-8-algebraic-identities [1] Grade 8 Algebraic Identities For more such worksheets visit www.edugain.com Answer t he quest ions (1) If, f ind the value of. (2) If 3(p 2 + q 2 + r 2 ) = (p + q + r) 2, f ind the value of p - 2q + r. (3) If xy = 9, and 2x - 5y = 9, f ind value of 4x 2 + 25y 2. (4) Solve the f ollowing using standard identities A) 42 2 B) 2.9 2 C) 299 2 D) 81 2 Choose correct answer(s) f rom given choice (5) Solve the f ollowing using the standard identity (x + a) (x + b) = x 2 + (a + b)x + ab 998 1003 a. 1001004 b. 1000994 c. 1000980 d. 1000990 (6) If, f ind the value of x 2 - y 2. a. 0 b. 9 3 c. 8 3 d. 3 (7) There are two numbers such that sum of the numbers is 12 and the sum of their squares is 104. Find their product. a. 20 b. 19 c. 25 d. 28 (8) If = 5, f ind the value of. a. 3 b. 7 c. 6 d. 2 (9) Find the value of (6.968)2 - (3.032) 2 using standard identities. a. 1 b. 20 c. 100 d. 10
(10) Solve the f ollowing using the standard identity (a+b) (a-b) = a 2 - b 2 1007 993 a. 999956 b. 999969 ID : ae-8-algebraic-identities [2] c. 999951 d. 999941 (11) Solve the f ollowing using the standard identity a 2 - b 2 = (a+b) (a-b) 84 2-16 2 a. 6783 b. 6814 c. 6800 d. 6796 (12) Find the value of (433.46)2 - (363.46) 2 using standard identities. a. 7 b. 70 c. 700 d. 140 (13) If (p - 1) 2 + (q - 5) 2 + (r - 5) 2 + (s - 3) 2 + (t - 2) 2 = 0, f ind the value of p + q + r + s + t. a. 32 b. 134 c. 25 d. 16 (14) Simplif y (2xy - 3yz) 2 + 12xy 2 z a. 4x 2 y 2 + 9y 2 z 2 - xy 2 z b. 4x 2 y 2 + 9y 2 z 2 c. 4x 2 y 2 + 9y 2 z 2 + xy 2 z d. 9x 2 y 2 + 4y 2 z 2 Fill in the blanks (15) If x 2 + y 2 = 26 and xy = 5, the value of 3(x + y) 2-4(x - y) 2 =. 2016 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : ae-8-algebraic-identities [3] (1) 2 (2) 0 (3) 261 It is given that: xy = 9 -----(1) It is also given that: 2x - 5y = 9 On squaring both sides we get: (2x - 5y) 2 = 81 (2x) 2 + (-5y) 2 + 2 (2x) (-5y) = 81...[Since, (a + b) 2 = a 2 + b 2 + 2ab] 4x 2 + 25y 2 + (-20)xy = 81 4x 2 + 25y 2 + (-20)(9) = 81...[From equation (1)] 4x 2 + 25y 2 = 261 Thus, the value of 4x 2 + 25y 2 is 261. (4) A) 1764 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 42 2 Now, 42 = 40 + 2 Theref ore, 42 2 = (40 + 2) 2 42 2 = 40 2 + 2 2 + (2 x 40 x 2) 42 2 = 1600 + 4 + 160 42 2 = 1764
B) 8.41 ID : ae-8-algebraic-identities [4] 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 2.9 2 Now, 2.9 = 3-0.1 Theref ore, 2.9 2 = (3-0.1) 2 2.9 2 = 3 2 + 0.1 2 - (2 x 3 x 0.1) 2.9 2 = 9 + 0.01-0.6 2.9 2 = 8.41 C) 89401 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 299 2 Now, 299 = 300-1 Theref ore, 299 2 = (300-1) 2 299 2 = 300 2 + 1 2 - (2 x 300 x 1) 299 2 = 90000 + 1-600 299 2 = 89401 D) 6561 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 81 2 Now, 81 = 80 + 1 Theref ore, 81 2 = (80 + 1) 2 81 2 = 80 2 + 1 2 + (2 x 80 x 1) 81 2 = 6400 + 1 + 160 81 2 = 6561
(5) b. 1000994 ID : ae-8-algebraic-identities [5] We have been asked to f ind the value of 998 1003 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 998. Two such simple numbers are 1000 and -2. Similarly, two simple numbers whose sum is 1003 are 1000 and 3. Thus, 998 1003 = { 1000 + (-2)} { 1000 + (3)} = 1000 2 + {(-2) + (3)} 1000 + (-2)(3)...[Using the identity (x + a) (x + b) = x 2 + (a + b)x + ab] = 1000000 + (1)(1000) + (-6) = 1000000 + (1000) + (-6) = 1000994 Theref ore, the result is 1000994. (6) c. 8 3 (7) a. 20 Let s assume the two numbers be x and y. It is given that, sum of their squares is 104. Theref ore, x 2 + y 2 = 104 -----(1) Also the sum of the numbers is 12. Theref ore, x + y = 12 On squaring both sides we get: (x + y) 2 = 144 x 2 + y 2 + 2xy = 144...[Since, (x + y) 2 = x 2 + y 2 + 2xy] 104 + 2xy = 144...[From eqution (1)] 2xy = 144-104 2xy = 40 xy = 40 2 xy = 20 Step 4 Thus, their product is 20.
(8) b. 7 ID : ae-8-algebraic-identities [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 = 5 + 2 = 7 Theref ore, the value of is 7. (9) d. 10 We have been asked to f ind the value of (6.968)2 - (3.032) 2 using standard identities. Now, (6.968) 2 - (3.032) 2 (6.968 + 3.032)(6.968-3.032) = b)(a - b) in the numerator] 10 = = 10 [By using the identity a 2 - b 2 = (a + Theref ore, the value of (6.968)2 - (3.032) 2 is 10.
(10) c. 999951 ID : ae-8-algebraic-identities [7] We have been asked to f ind the value of 1007 993 using the f ollowing identity: (a+b) (ab) = a 2 - b 2. Let us try to think of two numbers whose sum is 1007 and dif f erence is 993. Two such numbers are 1000 and 7. Thus, 1007 993 = (1000 + 7) (1000-7) = 1000 2-7 2 [Using the identity (a+b) (a-b) = a 2 - b 2 ] = 1000000-49 = 999951 Theref ore, the result is 999951. (11) c. 6800 We have been asked to f ind the value of 84 2-16 2 using the f ollowing identity: a 2 - b 2 = (a + b)(a - b). Applying the identity, we can write 84 2-16 2 as: (84 + 16)(84-16) = 100 68 = 6800 Theref ore, the result is 6800.
(12) b. 70 ID : ae-8-algebraic-identities [8] We have been asked to f ind the value of (433.46)2 - (363.46) 2 using standard identities. Now, (433.46) 2 - (363.46) 2 = (433.46 + 363.46)(433.46-363.46) [By using the identity a 2 - b 2 = (a + b)(a - b) in the numerator] 70 = = 70 Theref ore, the value of (433.46)2 - (363.46) 2 is 70. (13) d. 16 Given (p - 1) 2 + (q - 5) 2 + (r - 5) 2 + (s - 3) 2 + (t - 2) 2 = 0 It means the sum of (p - 1) 2, (q - 5) 2, (r - 5) 2, (s - 3) 2 and (t - 2) 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 (p - 1) 2, (q - 5) 2, (r - 5) 2, (s - 3) 2 and (t - 2) 2 can be zero only if all of them are also equal to zero. Now, (p - 1) 2 = 0 p - 1 = 0 p = 1 Similarly, q = 5, r = 5, s = 3, t = 2. Step 4 Thus, the value of p + q + r + s + t = 1 + 5 + 5 + 3 + 2 = 16
(14) b. 4x 2 y 2 + 9y 2 z 2 ID : ae-8-algebraic-identities [9] We know that (a - b) 2 = a 2 + b 2-2ab. Now, let us start simplif ying (2xy - 3yz) 2 + 12xy 2 z by applying the identity (a - b) 2 = a 2 + b 2-2ab to the part (2xy - 3yz) 2 : (2xy - 3yz) 2 + 12xy 2 z = (2xy) 2 + (3yz) 2-2(2xy)(3yz) + 12xy 2 z = 4x 2 y 2 + 9y 2 z 2-12xy 2 z + 12xy 2 z = 4x 2 y 2 + 9y 2 z 2 Thus, the given expression can be simplif ied as 4x 2 y 2 + 9y 2 z 2. (15) 44 It is given that, x 2 + y 2 = 26 and xy = 5 Now, 3(x + y) 2-4(x - y) 2 = 3(x 2 + y 2 + 2xy) - 4(x 2 + y 2-2xy) = 3x 2 + 3y 2 + 6xy - 4x 2-4y 2 + 8xy = -1x 2-1y 2 + 14xy = -1(x 2 + y 2 ) + 14xy = -1(26) + 14(5) = 44 Thus, the value of 3(x + y) 2-4(x - y) 2 is 44.