Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 482 · 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.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 1 and 32 (d) 2 and 16
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
4. 4.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 14 (d) 42
5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
6. 6.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
7. 7.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 85 (d) 169
8. 8.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 68 (d) 36
9. 9.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 4 (d) 64
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
11. 11.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 3625 (d) 4225
12. 12.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 7 (c) 109 (d) 169
13. 13.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 7235 (d) 8075
14. 14.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
15. 15.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
16. 16.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 169 (d) 145
17. 17.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)
18. 18.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 4225 (d) 3035
19. 19.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 3 and 6 (d) 2 and 9
20. 20.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 25 (d) 5
21. 21.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 400 (d) 304
22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
24. 24.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
25. 25.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 256 (d) 16
26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
27. 27.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 10 (d) 28
28. 28.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 141 (d) 121
29. 29.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 49 (d) 169
30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6401 (d) 6561
31. 31.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
32. 32.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 625 (d) 225
33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 41 (d) 81
34. 34.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
36. 36.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2611 (c) 2501 (d) 2401
37. 37.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 25 (d) 23
38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
40. 40.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23

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