Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 653 · 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 + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
3. 3.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
4. 4.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
5. 5.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 6 (d) 7
6. 6.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)
7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
8. 8.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
9. 9.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1225 (d) 1235
10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
11. 11.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409
12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1296 (d) 1306
13. 13.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
16. 16.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
18. 18.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
19. 19.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)
20. 20.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 100 (d) 148
21. 21.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 125 (d) 325
24. 24.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)
25. 25.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 4 and 7 (c) 2 and 14 (d) 1 and 10
26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
27. 27.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 2 and 10 (c) 3 and 9 (d) 1 and 27
28. 28.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 36 (d) 324
30. 30.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 52 (d) 76
31. 31.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 100 (d) 16
32. 32.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
33. 33.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
34. 34.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
35. 35.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
37. 37.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3969 (c) 3249 (d) 3979
38. 38.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
39. 39.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 4 (d) 16
40. 40.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 64 (c) 40 (d) 4

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