Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 649 · 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.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9811 (c) 9801 (d) 10201
2. 2.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2601 (d) 2411
3. 3.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
4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
5. 5.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
6. 6.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 529 (d) 299
7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
9. 9.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
10. 10.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
11. 11.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 198 (d) 144
12. 12.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
13. 13.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 1 and 4 (c) 2 and 3 (d) 2 and 4
14. 14.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
15. 15.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3969 (c) 3249 (d) 3591
16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
19. 19.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 7406 (d) 8836
20. 20.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 9 (d) 185
21. 21.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 229 (d) 169
22. 22.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
23. 23.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2304 (d) 2714
24. 24.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
25. 25.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
26. 26.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 3 and 8 (d) 2 and 12
27. 27.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 23 (c) 10 (d) 25
28. 28.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
29. 29.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 22 (d) 13
30. 30.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 9409 (d) 10609
31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
32. 32.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
33. 33.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 10 (c) 2 and 9 (d) 1 and 18
34. 34.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
35. 35.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 109 (d) 49
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
38. 38.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
39. 39.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6724 (d) 6084
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)

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