Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 72 · 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.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 65 (d) 93
2. 2.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2491 (c) 2809 (d) 2219
3. 3.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
5. 5.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
6. 6.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 17 (d) 19
7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
11. 11.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 1 and 9 (d) 2 and 8
12. 12.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
13. 13.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 4 (c) 16 (d) 100
14. 14.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
15. 15.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 31 (d) 11
16. 16.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
17. 17.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3035 (c) 3025 (d) 4225
18. 18.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 121 (d) 91
19. 19.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 21 (d) 19
20. 20.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
23. 23.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)
24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
25. 25.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
26. 26.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3844 (d) 3854
27. 27.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 208 (d) 144
29. 29.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
30. 30.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
31. 31.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
32. 32.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3364 (d) 3844
33. 33.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
34. 34.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
35. 35.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4225 (d) 4235
36. 36.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 229 (d) 409
37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
39. 39.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)

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