Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 772 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
2. 2.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 80 (d) 112
3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
4. 4.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3249 (d) 3969
5. 5.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 1 and 12 (d) 5 and 8
6. 6.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 1 and 12 (d) 2 and 18
7. 7.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 181 (d) 271
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
10. 10.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
11. 11.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 5776 (d) 6384
12. 12.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9
13. 13.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
14. 14.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3844 (d) 3596
15. 15.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 5184 (d) 4624
16. 16.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 19 (d) 25
17. 17.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
18. 18.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 100 (d) 10
19. 19.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101
20. 20.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
22. 22.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
24. 24.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 12 (d) 40
25. 25.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 9 (d) 8
26. 26.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 37 (d) 25
27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
28. 28.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 30 (d) 10
29. 29.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9614 (d) 9604
30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
31. 31.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 25 (d) 5
32. 32.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 2 and 6 (d) 1 and 15
33. 33.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 289 (d) 299
34. 34.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8474 (d) 8104
35. 35.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599
37. 37.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
38. 38.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
39. 39.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 2 and 8 (d) 3 and 7
40. 40.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)

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