Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 459 · 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 86²? (a) 7396 (b) 7406 (c) 8836 (d) 8084
2. 2.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
3. 3.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
4. 4.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
6. 6.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
8. 8.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
9. 9.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 65 (c) 97 (d) 81
10. 10.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 1 and 4 (d) 2 and 4
11. 11.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
12. 12.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
13. 13.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
14. 14.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
16. 16.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 109 (d) 49
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
18. 18.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 37 (d) 49
19. 19.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
20. 20.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)
21. 21.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1946 (d) 1936
22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
23. 23.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)
24. 24.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 7 (d) 11
25. 25.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2025 (d) 2525
26. 26.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 5 (d) 97
27. 27.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
28. 28.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 90 (d) 117
29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
30. 30.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 12 (c) 27 (d) 26
31. 31.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
32. 32.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
33. 33.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
34. 34.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)
35. 35.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 33 (c) 31 (d) 32
36. 36.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3364 (d) 3854
37. 37.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 324 (d) 16
38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)

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