Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 466 · 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.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 2 and 12 (d) 4 and 6
2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 5486 (d) 4916
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
4. 4.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 35 (d) 12
5. 5.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7
6. 6.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)
7. 7.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
8. 8.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 104 (c) 8 (d) 64
9. 9.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
10. 10.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
11. 11.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 247 (d) 170
12. 12.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 80 (d) 4
13. 13.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 9
14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625
15. 15.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
16. 16.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 27 (d) 3 and 9
17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
18. 18.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 421 (d) 289
19. 19.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4899 (d) 4761
20. 20.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)
21. 21.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1599 (d) 1521
22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
24. 24.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
25. 25.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 4489 (d) 5339
26. 26.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
27. 27.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 5 (d) 7
28. 28.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 2 and 20 (c) 1 and 40 (d) 1 and 12
29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
30. 30.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
31. 31.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 13 (d) 39
32. 32.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 85 (c) 127 (d) 169
33. 33.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 2 and 14 (d) 1 and 10
34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 64 (d) 16
35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
36. 36.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
37. 37.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 7 (d) 8
38. 38.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)
39. 39.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225
40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 9 (d) 185

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