Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 579 · 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 15²? (a) 235 (b) 375 (c) 225 (d) 625
2. 2.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
3. 3.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 5776 (d) 7056
4. 4.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
5. 5.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
6. 6.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225
7. 7.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
10. 10.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
11. 11.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
12. 12.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
14. 14.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 1 and 40 (d) 5 and 8
15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
16. 16.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 396 (d) 334
17. 17.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)
18. 18.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)
19. 19.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
20. 20.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 170 (d) 16
21. 21.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 2 (d) 16
22. 22.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
23. 23.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
24. 24.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
26. 26.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
27. 27.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 12 (d) 14
28. 28.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
29. 29.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 5184 (d) 4624
30. 30.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 196 (c) 130 (d) 8
31. 31.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 6 (d) 144
32. 32.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 11 (d) 12
33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 40 (d) 14
34. 34.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
36. 36.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 121 (c) 181 (d) 61
37. 37.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
38. 38.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 729 (d) 909
39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
40. 40.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)

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