Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 755 · 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 + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
3. 3.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 157 (d) 193
4. 4.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225
5. 5.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
6. 6.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 9 (d) 8
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
8. 8.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 7 and 9 (d) 1 and 15
9. 9.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 271 (d) 1
10. 10.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 8
11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
12. 12.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 132 (d) 68
13. 13.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)
14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
15. 15.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
16. 16.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
17. 17.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
18. 18.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
19. 19.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 81 (d) 41
21. 21.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
22. 22.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
23. 23.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
25. 25.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 7 (d) 11
26. 26.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 235 (d) 225
27. 27.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)
28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 43 (d) 13
29. 29.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
30. 30.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3596 (d) 3364
31. 31.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 5329 (d) 4489
32. 32.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)
33. 33.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 65 (d) 121
34. 34.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 266 (d) 384
36. 36.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 208 (d) 592
37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
39. 39.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 148 (d) 172

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