Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 86 · 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 + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
2. 2.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1521 (c) 1601 (d) 1691
3. 3.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
4. 4.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 2 and 16 (c) 4 and 8 (d) 1 and 32
5. 5.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
6. 6.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
7. 7.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 2025 (d) 1575
8. 8.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9614 (d) 9604
9. 9.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2704 (d) 2496
10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
11. 11.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
13. 13.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 625 (d) 635
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
15. 15.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
17. 17.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 136 (d) 196
18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
19. 19.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851
20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3491 (d) 3481
21. 21.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
22. 22.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
23. 23.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
24. 24.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
25. 25.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
29. 29.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
30. 30.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
31. 31.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
33. 33.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 141 (c) 101 (d) 111
34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
36. 36.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
37. 37.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
38. 38.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)

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