Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 895 · 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 = 4 and ab = 3, what is a² + b²? (a) 13 (b) 10 (c) 16 (d) 22
2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
3. 3.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
4. 4.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
5. 5.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 8 (d) 15
6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 4 (d) 2
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 4 (d) 16
8. 8.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
9. 9.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
10. 10.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 14 (d) 13
11. 11.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
12. 12.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 50 (d) 64
13. 13.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 625 (d) 1225
14. 14.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
15. 15.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 12 (d) 13
16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
17. 17.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 1 (d) 181
19. 19.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 122 (c) 100 (d) 144
20. 20.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 25 (c) 37 (d) 49
21. 21.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
22. 22.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 29 (d) 11
23. 23.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
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.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
26. 26.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3481 (d) 3731
27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
28. 28.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
29. 29.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 16 (d) 58
30. 30.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1625 (d) 1225
31. 31.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
32. 32.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
34. 34.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
36. 36.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 169 (d) 9
37. 37.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 4 (d) 324
39. 39.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
40. 40.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157

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