Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 792 · 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 18²? (a) 324 (b) 484 (c) 396 (d) 334
2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
3. 3.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 16 (d) 15
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
5. 5.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 5 (d) 25
6. 6.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)
7. 7.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
8. 8.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
9. 9.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 1 and 28 (d) 2 and 14
10. 10.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
11. 11.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 11025 (d) 9025
12. 12.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 144 (d) 122
13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
14. 14.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4225 (c) 4235 (d) 5625
15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
16. 16.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 225 (d) 3
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
18. 18.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
19. 19.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 106 (d) 286
20. 20.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)
21. 21.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 12 (d) 13
22. 22.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3601 (d) 3721
23. 23.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
24. 24.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 366 (c) 201 (d) 256
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.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
27. 27.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 4 and 5 (d) 2 and 10
28. 28.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 9 (d) 81
29. 29.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 157 (d) 223
30. 30.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
31. 31.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
32. 32.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 43 (d) 42
33. 33.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
34. 34.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
36. 36.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
37. 37.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
38. 38.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8836 (d) 8846
39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)

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