Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 489 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
3. 3.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 169 (c) 11 (d) 121
4. 4.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9604 (d) 9996
7. 7.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 91 (d) 1
8. 8.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
10. 10.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
11. 11.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 971 (c) 901 (d) 841
12. 12.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 4761 (d) 5041
13. 13.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 22 (d) 10
14. 14.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 1089 (c) 891 (d) 739
15. 15.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 9 (d) 81
16. 16.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 64 (c) 130 (d) 196
17. 17.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 4624 (d) 5184
18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
20. 20.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 41 (d) 61
21. 21.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
23. 23.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
24. 24.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 12
25. 25.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 5776 (d) 6384
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
28. 28.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
29. 29.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
31. 31.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
32. 32.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
34. 34.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
35. 35.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
37. 37.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277
38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
39. 39.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
40. 40.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14

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