Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 930 · 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.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 8 (d) 1 and 18
2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
3. 3.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 13 (d) 22
4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
5. 5.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)
6. 6.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2509 (c) 2809 (d) 2819
7. 7.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 81 (d) 117
8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
9. 9.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 376 (c) 136 (d) 256
10. 10.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1764 (c) 1774 (d) 1604
11. 11.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
12. 12.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3481 (c) 3599 (d) 3491
13. 13.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9025 (d) 9035
14. 14.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2499 (c) 2601 (d) 2401
15. 15.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 4 (d) 6
16. 16.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
17. 17.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 25 (d) 19
18. 18.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)
19. 19.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 94 (d) 49
20. 20.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 289 (d) 9
21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
22. 22.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 64 (c) 88 (d) 40
23. 23.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
24. 24.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
25. 25.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 29 (c) 39 (d) 49
26. 26.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
27. 27.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
28. 28.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 54 (d) 1 and 14
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
33. 33.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
34. 34.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2611 (d) 2401
35. 35.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 289 (d) 149
36. 36.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
37. 37.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 89 (d) 249
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
40. 40.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)

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