Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 435 · 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.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8084 (c) 8836 (d) 7396
2. 2.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
3. 3.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 45 (d) 14
4. 4.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
5. 5.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
7. 7.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
9. 9.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
10. 10.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 225 (d) 635
11. 11.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 289 (d) 429
12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
13. 13.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)
14. 14.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 256 (c) 64 (d) 160
15. 15.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
16. 16.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
17. 17.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 144 (d) 6
18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
19. 19.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
20. 20.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 7
21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
22. 22.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3721 (d) 3731
23. 23.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 27 (d) 13
24. 24.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1296 (c) 1616 (d) 1946
26. 26.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9614 (d) 9996
27. 27.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
28. 28.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
29. 29.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 10 (d) 3 and 8
30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 289 (d) 217
32. 32.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
33. 33.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 58 (c) 142 (d) 100
34. 34.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 4 and 8 (d) 2 and 16
35. 35.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
36. 36.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 1 and 18 (d) 3 and 6
37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
38. 38.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 2 and 9 (d) 3 and 6
39. 39.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 896 (d) 794
40. 40.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 144 (d) 2

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