Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 424 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
2. 2.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 196 (d) 244
3. 3.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
4. 4.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 47 (d) 48
5. 5.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3969 (c) 3591 (d) 3259
6. 6.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 196 (c) 292 (d) 148
7. 7.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
8. 8.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
9. 9.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 34 (c) 4 (d) 64
10. 10.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 9604 (d) 10414
11. 11.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 31 (d) 36
12. 12.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
13. 13.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 35 (d) 11
14. 14.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9614 (d) 9604
15. 15.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
16. 16.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401
17. 17.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
18. 18.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 109 (c) 81 (d) 53
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
20. 20.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 15 (d) 9
21. 21.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3721 (d) 3601
22. 22.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
23. 23.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
24. 24.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 1 and 10 (d) 4 and 7
25. 25.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
26. 26.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
27. 27.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 241 (d) 97
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
30. 30.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 73 (c) 121 (d) 25
31. 31.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 121 (d) 169
32. 32.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816
33. 33.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 576 (d) 256
34. 34.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
35. 35.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
36. 36.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
38. 38.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
39. 39.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 180 (d) 36
40. 40.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)

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