Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 367 · 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 = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
2. 2.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
3. 3.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 136 (d) 4
4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
7. 7.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 9216 (d) 10016
8. 8.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
9. 9.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 1 and 48 (d) 6 and 8
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
11. 11.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 1 and 28 (d) 2 and 14
12. 12.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
15. 15.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)
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
17. 17.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 9 (d) 20
18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
19. 19.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 97 (d) 169
20. 20.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 7 (d) 8
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 10816 (d) 9216
23. 23.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
25. 25.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
26. 26.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 49 (c) 7 (d) 225
27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
28. 28.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
30. 30.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 382 (d) 130
31. 31.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 5329 (d) 4499
32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 256 (d) 196
35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
37. 37.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 100 (d) 10
38. 38.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
39. 39.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 80 (c) 144 (d) 4
40. 40.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8

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