Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 87 · 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 + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
2. 2.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
3. 3.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 21 (d) 9
4. 4.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 122 (c) 100 (d) 144
5. 5.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 121 (d) 73
6. 6.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
7. 7.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)
8. 8.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 6 and 7 (d) 2 and 21
9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
10. 10.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9984 (d) 9216
11. 11.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
12. 12.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
13. 13.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
14. 14.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1306 (d) 1296
15. 15.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 4 (d) 256
16. 16.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 12 (d) 11
17. 17.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 4891 (d) 5329
18. 18.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
19. 19.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1936 (c) 1616 (d) 1946
20. 20.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
21. 21.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
22. 22.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
23. 23.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
24. 24.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
25. 25.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 10 (d) 100
26. 26.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
27. 27.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2035 (d) 2475
28. 28.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
29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 361 (d) 441
30. 30.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 12 (d) 8
31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
32. 32.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
33. 33.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
34. 34.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
35. 35.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)
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
37. 37.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
38. 38.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)
39. 39.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 19 (c) 9 (d) 18
40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)

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