Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 114 · 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 = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
2. 2.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
3. 3.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 421 (d) 157
4. 4.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
5. 5.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 9 (d) 11
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
7. 7.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
10. 10.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 484 (d) 324
11. 11.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8836 (d) 8116
12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
13. 13.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 9 (d) 8
14. 14.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
16. 16.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
19. 19.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 9614 (d) 10404
20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2025 (d) 2475
22. 22.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 11035 (d) 9025
23. 23.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
24. 24.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3969 (d) 3591
25. 25.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
26. 26.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
27. 27.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 116 (d) 36
28. 28.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
29. 29.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 49 (d) 169
30. 30.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 79 (d) 100
31. 31.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 14 (c) 40 (d) 12
32. 32.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
33. 33.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 13 (d) 10
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
35. 35.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 4 (d) 2
36. 36.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 1 and 6 (d) 3 and 4
37. 37.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 271 (d) 181
38. 38.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9801 (d) 9811
39. 39.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
40. 40.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 68 (c) 84 (d) 100

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