Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 675 · 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 = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 180 (d) 6
2. 2.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 7225 (d) 5625
3. 3.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 6 (d) 7
4. 4.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
5. 5.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
6. 6.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 3 (d) 17
7. 7.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 256 (d) 576
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
9. 9.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
10. 10.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 11 (d) 24
11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
12. 12.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 3 (d) 361
13. 13.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
14. 14.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 14 (d) 15
15. 15.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
16. 16.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
17. 17.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 6 (d) 146
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
20. 20.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
21. 21.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
22. 22.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 2 and 18 (d) 1 and 12
23. 23.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
25. 25.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
26. 26.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
27. 27.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
28. 28.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 10 (d) 1 and 28
29. 29.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
30. 30.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
32. 32.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
33. 33.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 13 (d) 11
34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6084 (d) 6094
35. 35.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 64 (d) 62
36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
37. 37.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
38. 38.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
40. 40.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3481 (d) 3731

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