Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 556 · 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.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)
2. 2.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
4. 4.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10816 (d) 10826
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
6. 6.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
7. 7.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 324 (d) 252
8. 8.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
9. 9.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 1 and 14 (d) 6 and 9
10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
11. 11.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3491 (d) 3481
12. 12.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 25 (d) 5
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
14. 14.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 104 (d) 144
15. 15.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 3 (d) 9
16. 16.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
17. 17.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 69 (d) 29
18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
20. 20.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4096 (d) 4106
21. 21.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
22. 22.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
24. 24.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 1 (d) 61
25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6396 (d) 6094
26. 26.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
28. 28.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 17 (c) 33 (d) 21
30. 30.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
31. 31.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
32. 32.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9984 (d) 9226
33. 33.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
34. 34.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7921 (d) 7931
35. 35.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
36. 36.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 11 (c) 121 (d) 145
37. 37.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 1 (d) 91
38. 38.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
39. 39.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
40. 40.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40

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