Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 709 · 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.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 169 (c) 121 (d) 11
2. 2.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3259 (d) 3249
3. 3.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 3025 (d) 2025
4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
5. 5.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 5041 (d) 4901
6. 6.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
7. 7.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 36 (d) 34
8. 8.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 4 (d) 324
9. 9.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4235 (d) 4875
10. 10.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 4925 (d) 5635
11. 11.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7931 (d) 7921
12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 7225 (d) 5625
13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
14. 14.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 61 (d) 1
15. 15.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 11 (d) 13
16. 16.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
18. 18.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
19. 19.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 529 (d) 409
20. 20.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
21. 21.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3844 (d) 3364
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
24. 24.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 6384 (d) 5776
25. 25.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 384 (c) 576 (d) 266
28. 28.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 9 (d) 10
29. 29.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
30. 30.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 13 (d) 1
31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 421 (d) 157
32. 32.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
33. 33.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 9 (d) 11
34. 34.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3364 (d) 3844
35. 35.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
36. 36.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 117
37. 37.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
38. 38.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 36 (d) 1 and 12
39. 39.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
40. 40.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884

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