Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 177 · 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 64²? (a) 4096 (b) 4106 (c) 3616 (d) 3136
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 361 (d) 441
3. 3.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 229 (d) 409
4. 4.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 52 (d) 100
5. 5.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 64 (d) 16
6. 6.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7396 (c) 8836 (d) 7406
7. 7.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 166 (c) 122 (d) 133
8. 8.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 6
9. 9.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
10. 10.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 6 (d) 7
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
12. 12.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
14. 14.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 9
15. 15.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
17. 17.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 1 and 27 (d) 2 and 10
18. 18.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 10 (d) 21
19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
20. 20.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
21. 21.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 5929 (d) 6889
23. 23.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 144 (d) 214
24. 24.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
25. 25.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 529 (d) 277
26. 26.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 64 (d) 4
27. 27.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 121 (d) 61
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
29. 29.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
30. 30.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
31. 31.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 10 (d) 11
32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
33. 33.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 899 (d) 841
34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
36. 36.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 58 (d) 142
37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
39. 39.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
40. 40.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)

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