Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 481 · 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 − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
4. 4.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
5. 5.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 163 (d) 196
6. 6.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35
7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
10. 10.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2704 (d) 2714
11. 11.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
12. 12.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
13. 13.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 371 (d) 441
14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625
15. 15.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4225 (d) 4235
16. 16.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 7 (d) 65
17. 17.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 8
18. 18.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 10
19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
20. 20.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 35 (d) 1 and 11
21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
22. 22.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
23. 23.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
24. 24.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
25. 25.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 45 (d) 63
26. 26.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
27. 27.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 49 (d) 73
28. 28.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 13 (d) 14
29. 29.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17
30. 30.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 52
31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 7225 (d) 8125
32. 32.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
33. 33.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
34. 34.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 11 (d) 28
35. 35.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
36. 36.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 13 (d) 25
37. 37.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3035 (d) 3025
39. 39.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 49 (c) 39 (d) 29
40. 40.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125

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