Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 728 · 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.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 10201 (d) 9999
2. 2.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
3. 3.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3025 (d) 3625
4. 4.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 6 (c) 144 (d) 36
5. 5.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 851 (d) 841
6. 6.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 52 (d) 100
7. 7.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
9. 9.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
10. 10.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 7225 (d) 9025
11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
13. 13.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
14. 14.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 2 (d) 4
15. 15.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2475 (d) 2025
16. 16.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
17. 17.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1936 (c) 1616 (d) 1946
18. 18.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 7225 (d) 9035
21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
22. 22.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 276 (d) 156
23. 23.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 6 (d) 100
24. 24.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
25. 25.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 4096 (d) 3146
26. 26.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
28. 28.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
29. 29.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 36 (d) 35
30. 30.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 169 (c) 109 (d) 229
31. 31.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
32. 32.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)
33. 33.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)
34. 34.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 12 (d) 11
35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
36. 36.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 8 (d) 64
37. 37.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
39. 39.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
40. 40.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)

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