Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 878 · 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 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
2. 2.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 9 (d) 8
3. 3.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
4. 4.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
5. 5.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 256 (d) 136
6. 6.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
7. 7.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 198 (d) 117
8. 8.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
9. 9.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
11. 11.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 81 (d) 121
12. 12.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 9419 (d) 10609
13. 13.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 229 (d) 169
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
15. 15.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
16. 16.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 76 (d) 148
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
18. 18.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 175 (c) 225 (d) 125
19. 19.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 148 (d) 100
20. 20.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 45 (d) 14
21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
23. 23.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
24. 24.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 13 (d) 12
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
27. 27.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6084 (d) 6724
28. 28.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 144 (c) 64 (d) 104
29. 29.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)
30. 30.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 81 (d) 117
31. 31.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 5 (d) 81
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9801 (d) 9811
33. 33.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 6 (d) 90
37. 37.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2025 (c) 3025 (d) 2035
38. 38.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 10 (d) 12
39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 121 (d) 181
40. 40.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2714 (d) 2704

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad