Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 992 · 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 a² + b²? (a) 61 (b) 91 (c) 121 (d) 181
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
3. 3.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
6. 6.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 50 (c) 78 (d) 57
7. 7.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 64 (c) 160 (d) 256
8. 8.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
9. 9.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 7406 (c) 8084 (d) 8836
10. 10.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3025 (d) 3035
11. 11.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 109 (c) 169 (d) 49
12. 12.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
13. 13.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
14. 14.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
17. 17.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 149 (d) 289
18. 18.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 256 (d) 193
19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
20. 20.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
21. 21.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
22. 22.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
23. 23.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 69 (d) 39
24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
25. 25.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)
26. 26.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
27. 27.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 6889 (d) 5929
28. 28.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
29. 29.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 324 (d) 484
30. 30.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
31. 31.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561
32. 32.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 12 (d) 1 and 42
33. 33.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
35. 35.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
36. 36.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
38. 38.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
39. 39.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
40. 40.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 58 (d) 79

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