Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 290 · 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.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
2. 2.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 31 (d) 26
3. 3.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
5. 5.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
6. 6.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)
7. 7.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
8. 8.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 20 (d) 8
9. 9.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 137 (c) 225 (d) 313
10. 10.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
11. 11.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 58 (c) 4 (d) 16
12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
13. 13.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 9025 (d) 8125
14. 14.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 103 (d) 85
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
16. 16.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 11
17. 17.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 229 (c) 289 (d) 409
18. 18.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
19. 19.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
20. 20.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
22. 22.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
24. 24.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
25. 25.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
27. 27.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 635 (d) 875
28. 28.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 113 (d) 225
29. 29.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 101 (d) 121
30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
31. 31.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 6 (d) 9
32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
33. 33.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
35. 35.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)
36. 36.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 6416 (d) 7066
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
38. 38.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 81 (c) 53 (d) 25
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 148 (d) 244

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