Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 700 · 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 + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
3. 3.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)
4. 4.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 10816 (d) 9216
5. 5.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
6. 6.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
7. 7.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 1 and 6 (c) 2 and 6 (d) 3 and 4
8. 8.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 34 (d) 64
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
12. 12.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
13. 13.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 9 (d) 18
14. 14.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 4 (d) 2
15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
16. 16.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 4 (c) 20 (d) 2
17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
18. 18.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
19. 19.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
20. 20.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 2 and 16 (d) 1 and 32
21. 21.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
23. 23.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
24. 24.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)
25. 25.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
27. 27.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5635 (d) 5625
28. 28.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
29. 29.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 169 (d) 9
30. 30.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 5 (d) 6
31. 31.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 2 and 14 (d) 1 and 10
32. 32.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
33. 33.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
35. 35.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
36. 36.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 256 (d) 4
37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2401 (d) 2411
38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
39. 39.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7931 (d) 7921
40. 40.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409

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