Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 591 · 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 is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
2. 2.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 15 (d) 14
3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 81 (d) 121
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
5. 5.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1849 (d) 1379
6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
7. 7.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)
8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
9. 9.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)
10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
11. 11.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247
12. 12.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 208 (d) 256
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
14. 14.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
16. 16.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 25 (d) 23
17. 17.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
19. 19.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
20. 20.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
21. 21.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
22. 22.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 97 (c) 241 (d) 169
23. 23.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2499 (d) 2411
24. 24.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
25. 25.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
27. 27.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
28. 28.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
29. 29.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
30. 30.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4761 (d) 4899
31. 31.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
32. 32.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 13 (d) 35
34. 34.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 16 (d) 10
35. 35.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 18 (d) 10
36. 36.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9
37. 37.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
38. 38.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 169 (d) 9
39. 39.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 10 (d) 1 and 24
40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)

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