Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 973 · 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 = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
3. 3.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10826 (d) 10016
4. 4.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)
5. 5.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
6. 6.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 125 (c) 225 (d) 5
7. 7.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
10. 10.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 49 (c) 85 (d) 7
11. 11.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 196 (d) 100
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
13. 13.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 8096 (d) 7744
14. 14.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
16. 16.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8281 (c) 8099 (d) 7921
17. 17.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
18. 18.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)
19. 19.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3609 (d) 3969
20. 20.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 6 (c) 1 and 10 (d) 2 and 5
21. 21.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2496 (c) 2314 (d) 2704
22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 324 (d) 484
23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
24. 24.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
25. 25.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 4 (c) 58 (d) 100
26. 26.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 5329 (d) 4499
27. 27.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1681 (d) 1521
28. 28.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 4 (d) 34
29. 29.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 425 (d) 225
30. 30.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
31. 31.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
32. 32.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 7066 (d) 5776
33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
34. 34.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 9 (d) 121
35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 794 (d) 1024
36. 36.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
37. 37.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
38. 38.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
39. 39.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 35 (d) 5 and 7
40. 40.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 13 (c) 3 and 15 (d) 1 and 45

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