Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 620 · 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 − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
2. 2.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
3. 3.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
4. 4.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 333 (d) 117
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
6. 6.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 97 (d) 169
7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 4925 (d) 5635
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
9. 9.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
10. 10.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
11. 11.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4761 (d) 4771
12. 12.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 85 (d) 157
13. 13.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4106 (d) 4096
14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 9025 (d) 11025
15. 15.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 256 (d) 352
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
17. 17.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
18. 18.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
19. 19.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 16 (d) 256
20. 20.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)
21. 21.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
22. 22.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 592 (d) 304
23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
25. 25.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
26. 26.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 153 (d) 189
27. 27.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4634 (d) 4896
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
29. 29.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 4904 (d) 5184
30. 30.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 7066 (d) 5776
31. 31.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
33. 33.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2304 (d) 2704
34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
36. 36.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 15 (d) 9
37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
39. 39.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
40. 40.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991

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