Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 734 · 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.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 34 (d) 64
2. 2.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 50 (d) 78
3. 3.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 15 (d) 3 and 5
4. 4.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
5. 5.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
6. 6.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
10. 10.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
12. 12.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
13. 13.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
14. 14.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
16. 16.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
17. 17.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 117 (d) 81
18. 18.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 225 (d) 113
19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
20. 20.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3721 (d) 3601
21. 21.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3491 (d) 3721
22. 22.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 429 (d) 219
23. 23.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
25. 25.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 9 (d) 169
26. 26.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
27. 27.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
28. 28.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 17 (d) 25
29. 29.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 29 (d) 31
30. 30.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 36 (d) 14
31. 31.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
33. 33.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
34. 34.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 225 (d) 9
35. 35.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 3 and 9 (c) 2 and 10 (d) 1 and 27
36. 36.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4916 (c) 4356 (d) 5476
37. 37.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271

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