Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 576 · 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 sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 9
2. 2.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 122 (c) 10 (d) 144
3. 3.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5184 (d) 5194
4. 4.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
5. 5.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)
6. 6.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 164 (d) 324
7. 7.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 1 (d) 127
8. 8.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851
9. 9.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
10. 10.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 4 and 5 (d) 1 and 20
11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
13. 13.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
14. 14.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 676 (d) 686
15. 15.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
16. 16.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 11 (d) 12
17. 17.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
18. 18.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 36 (d) 35
19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
20. 20.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
22. 22.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
23. 23.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
24. 24.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
26. 26.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 11 (c) 19 (d) 18
27. 27.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
28. 28.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 6 (d) 36
29. 29.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 9 (d) 12
30. 30.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 170 (d) 16
31. 31.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
32. 32.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
33. 33.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
35. 35.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 10009 (d) 9409
39. 39.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 73 (d) 37
40. 40.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3844 (d) 3854

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