Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 828 · 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.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 4 and 5 (c) 1 and 20 (d) 2 and 10
2. 2.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 208 (c) 256 (d) 352
3. 3.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 10 (d) 9
4. 4.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 256 (d) 4
5. 5.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
6. 6.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 7 (d) 1 and 12
7. 7.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
8. 8.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 9 (c) 7 (d) 5
9. 9.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 1 and 8 (c) 2 and 5 (d) 2 and 4
10. 10.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3374 (d) 3596
11. 11.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
12. 12.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 52 (d) 4
13. 13.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 10001 (d) 9801
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
15. 15.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 130 (d) 262
16. 16.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2504 (d) 2714
17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
18. 18.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 4 (c) 16 (d) 196
19. 19.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 112 (d) 80
20. 20.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
22. 22.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
23. 23.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 24 (d) 11
24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
25. 25.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
26. 26.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 8 (d) 10
27. 27.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
28. 28.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
29. 29.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
30. 30.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
31. 31.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9025 (d) 9035
32. 32.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
33. 33.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)
34. 34.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 247 (d) 170
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
37. 37.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
38. 38.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 2525 (d) 3035
39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
40. 40.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 9 (d) 17

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