Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 191 · 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 + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
2. 2.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
3. 3.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3249 (d) 3969
4. 4.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)
5. 5.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 4499 (d) 5329
6. 6.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4901 (d) 4761
7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
8. 8.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 1 (d) 169
9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
10. 10.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
11. 11.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 1 and 13 (d) 6 and 8
13. 13.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
14. 14.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
17. 17.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 40 (c) 1 and 12 (d) 2 and 20
18. 18.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 416 (d) 586
19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
20. 20.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 54 (c) 1 and 14 (d) 2 and 27
21. 21.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 90 (d) 144
22. 22.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
24. 24.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 4634 (d) 5184
26. 26.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
29. 29.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 8 (d) 64
30. 30.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
31. 31.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
32. 32.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
33. 33.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 5 (d) 49
34. 34.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
35. 35.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
37. 37.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
38. 38.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)
39. 39.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 36 (d) 324
40. 40.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 14 (d) 13

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