Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 302 · 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 − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
3. 3.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 36 (d) 6
4. 4.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
5. 5.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
9. 9.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 16 (d) 58
10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
12. 12.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3844 (d) 3596
13. 13.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
14. 14.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 5929 (d) 6391
15. 15.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
16. 16.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1601 (d) 1691
17. 17.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
19. 19.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 196 (d) 130
20. 20.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 10 (c) 12 (d) 24
21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
22. 22.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 325 (d) 175
24. 24.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 2 and 6 (d) 3 and 4
25. 25.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
26. 26.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 8 (d) 7
27. 27.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
28. 28.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 9409 (d) 10609
29. 29.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 7 (d) 11
30. 30.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
31. 31.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
32. 32.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
35. 35.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101
36. 36.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
37. 37.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 44 (d) 14
38. 38.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 193 (d) 157
39. 39.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 15 (d) 14
40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 409 (c) 169 (d) 289

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