Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 510 · 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 + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
2. 2.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 1 and 7 (d) 3 and 4
3. 3.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
4. 4.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
6. 6.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 15 (d) 8
7. 7.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9991 (c) 9409 (d) 9419
8. 8.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
10. 10.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3616 (d) 3136
11. 11.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)
12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 4366 (d) 5476
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
14. 14.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 18 (d) 9
15. 15.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
16. 16.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 14 (d) 12
17. 17.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7921 (d) 7931
18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
19. 19.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 6 (d) 144
20. 20.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 1 and 40 (d) 2 and 20
21. 21.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
22. 22.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
24. 24.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)
25. 25.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)
26. 26.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
27. 27.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 43 (c) 49 (d) 61
28. 28.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
29. 29.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 36 (c) 256 (d) 6
30. 30.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
31. 31.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 17 (d) 25
32. 32.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 657 (d) 333
33. 33.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 11 (d) 13
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
35. 35.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1225 (d) 1625
36. 36.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
37. 37.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816
38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
39. 39.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404

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