Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 770 · 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 − 15x + 54 = 0? (a) 15 (b) 54 (c) 55 (d) 53
2. 2.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 20 (d) 2
3. 3.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
4. 4.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 225 (d) 635
5. 5.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
6. 6.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 12 (d) 11
7. 7.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
10. 10.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 289 (d) 223
11. 11.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
13. 13.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 451 (d) 401
14. 14.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
16. 16.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 10 (c) 100 (d) 122
17. 17.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
19. 19.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1296 (c) 1936 (d) 1306
20. 20.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
21. 21.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
22. 22.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 12 (c) 14 (d) 36
23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
24. 24.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
25. 25.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 10 (d) 1 and 24
26. 26.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
27. 27.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)
28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
29. 29.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8109 (d) 8659
30. 30.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 9 (c) 3 (d) 81
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
32. 32.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 100 (d) 68
33. 33.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
34. 34.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 22 (d) 13
35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
36. 36.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10826 (c) 9216 (d) 10016
37. 37.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 1 and 13 (d) 5 and 9
38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 31 (d) 29
39. 39.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 20 (d) 28
40. 40.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 324 (c) 180 (d) 468

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