Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 413 · 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 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601
2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 289 (d) 145
3. 3.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
4. 4.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
6. 6.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
8. 8.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
10. 10.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3025 (d) 3625
11. 11.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2499 (d) 2401
12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
13. 13.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
14. 14.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 144 (d) 109
15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 400 (d) 598
17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
18. 18.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7056 (c) 7066 (d) 6416
19. 19.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 4 (d) 164
20. 20.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 6 and 8 (d) 2 and 24
22. 22.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
24. 24.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 100 (d) 4
25. 25.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 73 (c) 121 (d) 25
26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
27. 27.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
29. 29.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
30. 30.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 8 (d) 64
31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
32. 32.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
33. 33.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
34. 34.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7569 (c) 8649 (d) 7579
35. 35.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
36. 36.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 11 (d) 169
37. 37.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 7 (d) 11
38. 38.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
40. 40.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 225 (d) 81

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