Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 969 · 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 + 1)(x + 45) (d) (x − 5)(x − 9)
2. 2.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
3. 3.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
4. 4.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 11 (d) 28
5. 5.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 576 (d) 586
6. 6.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
7. 7.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10826 (d) 10016
8. 8.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)
9. 9.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
10. 10.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
11. 11.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 25 (d) 5
12. 12.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 16 (d) 170
13. 13.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
14. 14.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
15. 15.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 11 (d) 30
16. 16.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 2025 (d) 1225
17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
18. 18.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
20. 20.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
21. 21.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
22. 22.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
23. 23.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
24. 24.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
25. 25.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 1 and 11 (d) 5 and 7
26. 26.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
27. 27.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 5 (d) 4
28. 28.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
29. 29.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2916 (d) 2926
30. 30.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 247 (d) 478
31. 31.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
32. 32.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
34. 34.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2025 (d) 2035
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
38. 38.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9025 (d) 9035
39. 39.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)
40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 244 (c) 196 (d) 172

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