Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 866 · 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 + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
2. 2.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
3. 3.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 12 (d) 1 and 42
4. 4.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 117 (c) 63 (d) 81
5. 5.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 4 and 5 (d) 1 and 8
6. 6.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2475 (d) 2035
7. 7.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 4 (d) 5
8. 8.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
9. 9.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
10. 10.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
12. 12.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
13. 13.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
14. 14.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 81 (d) 225
15. 15.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 17 (d) 19
16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
18. 18.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
19. 19.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
21. 21.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
22. 22.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
23. 23.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
24. 24.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
25. 25.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
26. 26.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3025 (d) 3625
27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 112 (d) 144
29. 29.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 6391 (d) 5939
30. 30.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 35 (d) 12
31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
32. 32.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
34. 34.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
35. 35.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
36. 36.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 35 (c) 37 (d) 13
37. 37.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 5786 (d) 7056
38. 38.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)
39. 39.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 7 (d) 12
40. 40.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 196

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