Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 467 · 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 is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 14 (c) 13 (d) 9
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
3. 3.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
4. 4.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7
5. 5.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414
6. 6.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 6 (d) 5
7. 7.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
8. 8.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6084 (d) 6404
9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
10. 10.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
11. 11.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 1 (d) 225
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
13. 13.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2035 (d) 2025
14. 14.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
15. 15.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 1 (d) 81
16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
17. 17.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 85 (d) 121
18. 18.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 196
19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
20. 20.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
21. 21.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)
22. 22.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
23. 23.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 1 and 54 (d) 2 and 27
24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81
25. 25.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
26. 26.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)
27. 27.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
28. 28.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
29. 29.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 1 and 4 (d) 2 and 4
30. 30.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
31. 31.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
32. 32.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
33. 33.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
34. 34.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2304 (d) 2714
35. 35.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 9 (d) 81
36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
37. 37.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 3 (d) 185
38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
40. 40.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 7056 (d) 6416

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