Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 511 · 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 + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
2. 2.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 6 (c) 144 (d) 90
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
4. 4.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
5. 5.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
6. 6.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8464 (d) 8096
7. 7.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
8. 8.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 10 (d) 7
10. 10.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
11. 11.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 64 (d) 50
12. 12.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
13. 13.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
15. 15.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
16. 16.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 1225 (d) 2035
17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
18. 18.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 64 (d) 8
19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
20. 20.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
21. 21.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
22. 22.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1774 (d) 1764
23. 23.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1379 (d) 1369
24. 24.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3481 (d) 3491
25. 25.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
26. 26.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 94 (d) 49
27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
28. 28.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9996 (c) 9614 (d) 10404
29. 29.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 169 (d) 109
30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
31. 31.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 576 (d) 416
32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
34. 34.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 2 (d) 144
35. 35.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 196 (d) 4
36. 36.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
37. 37.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
38. 38.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
39. 39.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 100 (d) 148
40. 40.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)

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