Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 364 · 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 = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 112 (d) 80
2. 2.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2401 (d) 2611
3. 3.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)
4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
6. 6.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 2 and 14 (d) 1 and 28
7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
9. 9.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1296 (d) 1946
10. 10.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2116 (d) 2916
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
13. 13.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 100 (d) 91
14. 14.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
16. 16.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
17. 17.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 25 (c) 23 (d) 24
18. 18.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
19. 19.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
20. 20.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
21. 21.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
24. 24.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 2035 (d) 1225
25. 25.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
26. 26.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3854 (c) 3604 (d) 3364
27. 27.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 4 (c) 80 (d) 16
28. 28.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 1 and 15 (d) 3 and 5
29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 361 (d) 441
30. 30.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 4 (d) 36
31. 31.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 276 (d) 116
32. 32.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
33. 33.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10609 (d) 10009
34. 34.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 2 and 3 (d) 1 and 6
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
36. 36.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
37. 37.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
39. 39.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
40. 40.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 8 (d) 104

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