Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 968 · 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 + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
3. 3.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
5. 5.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
6. 6.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
7. 7.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 2 and 3 (d) 1 and 6
8. 8.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 10 (d) 18
9. 9.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225
10. 10.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 35 (d) 12
11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
12. 12.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 17 (c) 21 (d) 25
13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 289 (d) 149
16. 16.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
17. 17.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
18. 18.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 9
19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
20. 20.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
22. 22.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 384 (c) 266 (d) 576
23. 23.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
24. 24.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
25. 25.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
26. 26.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 35 (d) 34
27. 27.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
28. 28.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 3 and 9 (c) 1 and 27 (d) 1 and 11
29. 29.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
30. 30.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 2035 (d) 1225
31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
32. 32.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
33. 33.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
34. 34.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
35. 35.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 6 (c) 144 (d) 90
37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
38. 38.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
39. 39.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181

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