Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 419 · 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 + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
3. 3.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
4. 4.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
6. 6.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 9 (d) 10
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
8. 8.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
9. 9.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
10. 10.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 100 (d) 79
11. 11.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1454 (d) 1596
12. 12.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 1 and 45 (c) 3 and 15 (d) 5 and 9
13. 13.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 61 (d) 91
14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
15. 15.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 3 (c) 1 and 6 (d) 2 and 4
16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
20. 20.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2484 (d) 2116
21. 21.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
22. 22.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 2025 (d) 3025
23. 23.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 1 and 14 (d) 6 and 9
24. 24.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
25. 25.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)
26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
28. 28.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 25 (d) 225
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
30. 30.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 361 (d) 5
31. 31.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 68 (d) 100
32. 32.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 2025 (d) 1225
33. 33.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
35. 35.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2916 (c) 2516 (d) 2116
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
37. 37.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
38. 38.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
39. 39.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 4366 (d) 5476
40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)

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