Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 95 · 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 + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
2. 2.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 122 (d) 144
3. 3.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 84 (d) 132
4. 4.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 106 (c) 151 (d) 286
5. 5.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2509 (c) 2209 (d) 2819
6. 6.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3249 (d) 3609
8. 8.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 5 (c) 49 (d) 37
9. 9.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
12. 12.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
13. 13.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
14. 14.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 12 (d) 27
15. 15.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
16. 16.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 125 (c) 325 (d) 175
17. 17.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
18. 18.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 124 (d) 184
19. 19.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)
20. 20.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8649 (d) 8109
21. 21.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10016 (c) 9216 (d) 10826
22. 22.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
23. 23.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 12 (d) 14
24. 24.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 137 (c) 225 (d) 7
25. 25.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
26. 26.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 400 (d) 598
27. 27.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
28. 28.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 78 (c) 64 (d) 57
29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
30. 30.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
31. 31.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)
32. 32.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 121 (d) 169
33. 33.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 133 (d) 169
34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 256 (d) 196
35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
36. 36.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 794 (d) 1024
37. 37.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 100 (c) 68 (d) 36
38. 38.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 223 (d) 289
39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
40. 40.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 16

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