Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 693 · 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 + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
2. 2.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
3. 3.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
4. 4.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 2 and 4 (d) 1 and 8
5. 5.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4875 (d) 4225
6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
8. 8.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 14 (d) 47
9. 9.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 12 (d) 7
10. 10.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
11. 11.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 25 (d) 49
12. 12.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 202 (d) 400
13. 13.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)
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
15. 15.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
16. 16.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 163 (c) 196 (d) 262
17. 17.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7235 (d) 7225
18. 18.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 289 (d) 25
19. 19.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2475 (d) 2025
20. 20.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
21. 21.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 16 (d) 256
22. 22.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
23. 23.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 64 (d) 8
24. 24.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)
25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
26. 26.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 31 (d) 46
27. 27.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2209 (c) 2809 (d) 2819
28. 28.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 4916 (d) 5476
29. 29.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
31. 31.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 784 (d) 1024
32. 32.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 4904 (d) 5184
33. 33.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1859 (d) 1849
34. 34.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
35. 35.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
36. 36.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 145 (c) 169 (d) 11
37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 16 (d) 208
39. 39.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 409 (c) 169 (d) 289
40. 40.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 1 and 12 (d) 2 and 18

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