Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 429 · 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 = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 4 (d) 10
2. 2.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 25 (d) 5
3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
4. 4.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
5. 5.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6241 (d) 6571
6. 6.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
7. 7.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6561 (d) 6251
8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
11. 11.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 11 (d) 10
12. 12.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
13. 13.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 53 (c) 55 (d) 15
14. 14.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5339 (d) 5329
15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
16. 16.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32
17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
18. 18.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 57 (d) 55
19. 19.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 81 (c) 67 (d) 53
20. 20.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 20 (c) 36 (d) 4
21. 21.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
22. 22.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 169 (c) 11 (d) 145
23. 23.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)
24. 24.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 10
25. 25.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
26. 26.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
27. 27.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 13 (c) 15 (d) 14
28. 28.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)
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 180 (d) 6
30. 30.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 1 (c) 49 (d) 37
31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 2 (d) 164
32. 32.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9025 (d) 9035
33. 33.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
34. 34.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
36. 36.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 8104 (d) 7744
37. 37.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
38. 38.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 16 (d) 136
39. 39.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
40. 40.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521

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