Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 183 · 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 + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
2. 2.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 12 (d) 11
3. 3.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
4. 4.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 225 (c) 337 (d) 113
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7579 (d) 7569
7. 7.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
8. 8.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6724 (d) 6734
9. 9.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 169 (d) 85
10. 10.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
12. 12.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 64 (d) 34
13. 13.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
15. 15.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6561 (d) 6401
16. 16.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 2 and 5 (c) 3 and 4 (d) 1 and 6
17. 17.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
18. 18.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 189 (d) 297
19. 19.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
20. 20.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 4 (d) 256
21. 21.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
22. 22.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 25 (d) 53
23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 61 (d) 121
26. 26.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
27. 27.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)
28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
29. 29.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
30. 30.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3025 (c) 3575 (d) 3035
31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
32. 32.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 1 (d) 19
33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
34. 34.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 144 (d) 6
35. 35.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
36. 36.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27
37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
38. 38.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
39. 39.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 9 (d) 10
40. 40.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9

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