Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 658 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 68 (d) 84
2. 2.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 1 (d) 169
3. 3.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 73 (d) 25
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
5. 5.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 40 (d) 39
6. 6.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
8. 8.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 8 (d) 10
9. 9.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
10. 10.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 147 (d) 213
11. 11.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
12. 12.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 2 and 7 (c) 1 and 8 (d) 3 and 6
13. 13.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 97 (d) 169
14. 14.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
15. 15.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
16. 16.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
17. 17.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
18. 18.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251
19. 19.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 169 (c) 229 (d) 139
20. 20.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
22. 22.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 2 (d) 100
23. 23.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
24. 24.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 45 (d) 44
25. 25.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
26. 26.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 106 (d) 16
27. 27.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
28. 28.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
30. 30.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
32. 32.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 1225 (d) 625
33. 33.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
34. 34.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
35. 35.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5339 (c) 5329 (d) 4909
36. 36.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3616 (c) 4106 (d) 3136
37. 37.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 11 (d) 145
38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 400 (d) 301
39. 39.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
40. 40.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 116 (d) 196

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