Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 737 · 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 = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
3. 3.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
4. 4.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 2 (d) 4
5. 5.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)
6. 6.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
7. 7.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 3025 (d) 2475
8. 8.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
9. 9.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8099 (d) 8281
10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1859 (c) 1369 (d) 1609
11. 11.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 11 (d) 10
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
13. 13.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 5929 (d) 6889
14. 14.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
16. 16.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 299 (d) 289
17. 17.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 180 (d) 324
18. 18.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
22. 22.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 64 (d) 94
23. 23.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
24. 24.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
26. 26.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 5051 (d) 4901
27. 27.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 23 (c) 25 (d) 10
28. 28.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
29. 29.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
30. 30.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
31. 31.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 6 (d) 5
32. 32.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
33. 33.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2025 (d) 2035
34. 34.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 3 and 4 (d) 2 and 6
35. 35.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
36. 36.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
37. 37.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2401 (d) 2611
38. 38.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 52 (d) 4
39. 39.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)
40. 40.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 6384 (c) 5776 (d) 5786

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