Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 541 · 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.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
2. 2.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 4 and 6 (d) 2 and 12
3. 3.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)
4. 4.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
5. 5.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
6. 6.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
7. 7.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 11 (d) 12
8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
9. 9.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 144 (c) 104 (d) 64
10. 10.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
11. 11.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
12. 12.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 11025 (d) 9025
13. 13.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
14. 14.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 4 and 9 (d) 2 and 18
15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
17. 17.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 213 (d) 125
18. 18.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 12 (d) 11
19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
21. 21.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 3 and 6 (c) 1 and 8 (d) 1 and 14
22. 22.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 16 (d) 324
23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
25. 25.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 33 (c) 32 (d) 31
26. 26.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 9
27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
28. 28.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 46 (d) 44
29. 29.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
30. 30.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 196 (c) 151 (d) 286
31. 31.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 25 (c) 23 (d) 24
32. 32.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
33. 33.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 36 (c) 26 (d) 16
34. 34.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
35. 35.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)
36. 36.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 1024 (d) 794
37. 37.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 10 (d) 11
38. 38.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 169 (c) 5 (d) 97
39. 39.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 3 (d) 361

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