Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 306 · 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.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 19 (d) 17
2. 2.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 11 (c) 13 (d) 12
3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 361 (d) 5
4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
6. 6.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
7. 7.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
10. 10.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 5 (c) 97 (d) 25
11. 11.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2809 (d) 2491
12. 12.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
13. 13.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 3 and 4 (d) 1 and 12
14. 14.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 11 (d) 30
15. 15.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 11
16. 16.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 58 (d) 100
17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
18. 18.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 2 and 5 (d) 1 and 8
19. 19.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 169 (d) 409
20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
21. 21.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
22. 22.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 271 (d) 1
23. 23.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 16 (c) 144 (d) 80
24. 24.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 244 (c) 196 (d) 172
26. 26.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 175 (d) 225
27. 27.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 14 (d) 15
28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
29. 29.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 45 (d) 15
30. 30.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 529 (d) 361
31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
33. 33.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)
34. 34.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
36. 36.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 41 (d) 13
37. 37.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 49 (c) 25 (d) 37
38. 38.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
39. 39.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)
40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64

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