Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 654 · 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.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
3. 3.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 41 (c) 13 (d) 39
4. 4.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
6. 6.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2035 (d) 2475
7. 7.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
12. 12.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 4 (d) 256
13. 13.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9984 (d) 9216
14. 14.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
15. 15.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 6 (c) 90 (d) 36
16. 16.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 25 (d) 125
17. 17.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2509 (d) 2209
18. 18.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 262 (c) 196 (d) 163
19. 19.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 100 (d) 196
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
21. 21.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
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.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 9 (d) 225
24. 24.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
25. 25.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
26. 26.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 10201 (d) 9801
27. 27.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 13 (d) 12
28. 28.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)
29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
30. 30.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 149 (d) 9
31. 31.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
32. 32.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 277 (d) 361
33. 33.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 100 (d) 6
34. 34.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
35. 35.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
38. 38.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 91 (d) 121
39. 39.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 361 (d) 371
40. 40.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 10 (d) 11

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