Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 790 · 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 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
2. 2.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 19 (d) 25
3. 3.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
4. 4.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
6. 6.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 8836 (c) 8116 (d) 7396
8. 8.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 9 (d) 11
9. 9.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 729 (d) 739
10. 10.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1681 (d) 1521
11. 11.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 10 (d) 16
12. 12.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
13. 13.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 13 (d) 11
14. 14.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
15. 15.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 85 (d) 127
16. 16.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
17. 17.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
18. 18.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)
19. 19.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 324 (c) 396 (d) 334
20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
21. 21.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
22. 22.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
23. 23.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
24. 24.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
26. 26.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 5 (d) 169
27. 27.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
28. 28.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 6 (d) 36
29. 29.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 1225 (d) 625
30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
31. 31.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
32. 32.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 468 (d) 252
33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 5051 (d) 4761
34. 34.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
35. 35.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 4 and 5 (d) 1 and 8
36. 36.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)
37. 37.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
38. 38.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)
39. 39.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 324 (d) 6
40. 40.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 11 (c) 17 (d) 18

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