Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 433 · 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 − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
2. 2.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 36 (c) 46 (d) 31
3. 3.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)
4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
5. 5.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
6. 6.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3491 (d) 3599
7. 7.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
9. 9.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
10. 10.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 130 (d) 193
12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
14. 14.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
15. 15.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 36 (c) 6 (d) 196
16. 16.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 121 (d) 101
17. 17.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
18. 18.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
19. 19.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
21. 21.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 12 (d) 13
22. 22.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
25. 25.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 80 (d) 112
26. 26.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)
27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
28. 28.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)
29. 29.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
30. 30.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
32. 32.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 196 (c) 130 (d) 8
33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 14 (d) 13
34. 34.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
35. 35.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 9 (d) 8
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
39. 39.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
40. 40.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 2 and 16 (d) 1 and 32

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