Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 921 · 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 + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
3. 3.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
4. 4.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 196 (c) 106 (d) 4
5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
6. 6.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 875 (d) 1225
7. 7.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 39 (d) 40
8. 8.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
9. 9.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1616 (c) 1946 (d) 1296
10. 10.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3481 (d) 3491
11. 11.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 9 (d) 3
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 1 and 48 (d) 2 and 24
13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
16. 16.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
20. 20.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 34 (c) 36 (d) 12
21. 21.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 9025 (d) 11035
22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
25. 25.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
26. 26.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 170 (d) 478
27. 27.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 121 (c) 169 (d) 145
28. 28.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
29. 29.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
30. 30.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
31. 31.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
32. 32.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
33. 33.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
34. 34.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2491 (c) 2809 (d) 2209
35. 35.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
36. 36.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 794 (d) 784
37. 37.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 409 (d) 229
38. 38.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 67 (d) 109
39. 39.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
40. 40.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 9 (d) 8

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