Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 953 · 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 + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
2. 2.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 48 (d) 1 and 13
3. 3.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 9 (d) 11
4. 4.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
5. 5.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 277 (d) 361
6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
7. 7.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 45 (d) 15
8. 8.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 64 (d) 100
9. 9.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 9 (d) 45
10. 10.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
12. 12.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 8 (d) 6
13. 13.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 91 (c) 61 (d) 121
14. 14.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
15. 15.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
16. 16.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6084 (d) 6724
17. 17.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
18. 18.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 4 (d) 16
19. 19.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 10 (d) 11
20. 20.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
21. 21.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
22. 22.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 289 (d) 3
24. 24.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
25. 25.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 382 (d) 130
27. 27.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
28. 28.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
30. 30.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 175 (d) 325
31. 31.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
32. 32.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 12 (d) 18
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
35. 35.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 2035 (d) 1625
36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
38. 38.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 5 (c) 25 (d) 81
39. 39.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 7
40. 40.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971

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