Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 180 · 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 + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
3. 3.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7066 (d) 7056
4. 4.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
7. 7.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2525 (c) 2025 (d) 3025
8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
11. 11.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
12. 12.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
13. 13.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 7 (d) 9
14. 14.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 113 (d) 225
15. 15.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 10609 (c) 9409 (d) 9991
16. 16.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 169 (d) 229
17. 17.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)
18. 18.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
19. 19.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7754 (d) 7744
20. 20.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
21. 21.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
23. 23.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1521 (d) 1599
24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 81 (d) 121
25. 25.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 9 (d) 3
26. 26.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
27. 27.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
28. 28.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 10404 (d) 9996
29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
30. 30.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)
31. 31.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)
32. 32.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
33. 33.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 10 (d) 3 and 8
34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
36. 36.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 58 (c) 4 (d) 16
37. 37.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
38. 38.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 196 (d) 130
39. 39.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
40. 40.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14

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