Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 794 · 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 75²? (a) 6375 (b) 7225 (c) 5625 (d) 5635
2. 2.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 9 (d) 3
3. 3.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
4. 4.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
6. 6.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125
7. 7.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289
8. 8.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 7 (c) 8 (d) 10
10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 144 (d) 214
11. 11.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
12. 12.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 333 (d) 117
13. 13.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)
14. 14.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 7 (d) 49
15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
16. 16.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3609 (d) 3979
17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 13 (d) 12
18. 18.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 3025 (d) 2475
19. 19.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 65 (c) 7 (d) 81
20. 20.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
21. 21.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 28 (d) 52
22. 22.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225
23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
24. 24.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
25. 25.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 196 (d) 100
26. 26.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10001 (d) 10201
27. 27.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 27 (d) 28
28. 28.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 24 (d) 11
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
30. 30.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
31. 31.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 12 (d) 11
32. 32.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 13 (d) 14
33. 33.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
34. 34.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 2025 (d) 1225
35. 35.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 9 (d) 3
36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 170 (c) 247 (d) 324
37. 37.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 68 (d) 100
38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
39. 39.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2501 (d) 2601
40. 40.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)

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