Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 120 · 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.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 225 (d) 425
2. 2.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6401 (d) 6561
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
5. 5.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
7. 7.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
8. 8.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
11. 11.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 81 (d) 225
14. 14.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 9811 (d) 10201
15. 15.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
16. 16.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409
17. 17.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
19. 19.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6404 (d) 6724
20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 10 (d) 24
21. 21.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
23. 23.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 1 (d) 271
25. 25.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 144 (c) 124 (d) 184
26. 26.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 89 (c) 249 (d) 129
27. 27.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 289 (d) 7
28. 28.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 6 (c) 324 (d) 180
30. 30.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 160 (d) 352
31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
32. 32.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 11 (d) 12
33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25
34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 324 (d) 180
35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
36. 36.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
37. 37.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
38. 38.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2499 (d) 2411
39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
40. 40.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9

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