Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 251 · 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 + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
3. 3.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 2 and 14 (d) 4 and 7
4. 4.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
5. 5.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)
6. 6.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
7. 7.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
8. 8.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 36 (d) 116
9. 9.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 39 (d) 69
10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 1024 (d) 794
11. 11.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
12. 12.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 184 (d) 124
13. 13.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10609 (d) 10619
14. 14.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
15. 15.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 14 (d) 15
16. 16.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
17. 17.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 244 (d) 196
18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 2475 (d) 3025
22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
23. 23.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 324 (d) 4
24. 24.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 9 (d) 17
25. 25.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
26. 26.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 117
27. 27.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 37 (d) 49
28. 28.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
29. 29.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2491 (d) 2219
30. 30.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)
31. 31.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 136 (d) 4
32. 32.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 16 (c) 40 (d) 64
33. 33.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
34. 34.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
37. 37.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
38. 38.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
39. 39.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 3 (d) 121
40. 40.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 157 (d) 145

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