Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 139 · 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 product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 17 (d) 19
2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
3. 3.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 121 (d) 25
4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 9 (d) 17
5. 5.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 15 (d) 48
6. 6.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754
7. 7.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4225 (d) 4875
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
9. 9.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
10. 10.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 11 (c) 24 (d) 25
11. 11.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
12. 12.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9975 (c) 11025 (d) 9035
13. 13.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 25 (c) 37 (d) 49
14. 14.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 12 (d) 11
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
16. 16.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 1 and 12 (d) 5 and 8
17. 17.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 2 (c) 74 (d) 144
18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
19. 19.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
20. 20.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 29 (d) 28
21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
22. 22.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
24. 24.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
25. 25.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 25 (d) 13
26. 26.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)
27. 27.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 13 (d) 39
28. 28.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 10 (d) 18
29. 29.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2475 (d) 2035
30. 30.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 25 (d) 37
31. 31.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 5929 (c) 6409 (d) 6889
32. 32.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 196 (d) 172
33. 33.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 74 (c) 109 (d) 214
34. 34.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 4 (d) 2
35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
37. 37.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
38. 38.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 91 (c) 121 (d) 61
39. 39.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
40. 40.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13

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