Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 891 · 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 = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 49 (d) 73
2. 2.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 97 (d) 241
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.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
5. 5.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3854 (d) 3364
6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1625 (d) 1225
8. 8.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
10. 10.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2219 (c) 2809 (d) 2491
11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
15. 15.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 100 (d) 122
16. 16.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 127 (d) 169
17. 17.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
18. 18.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
19. 19.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4489 (c) 4499 (d) 4891
20. 20.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5329 (d) 5339
21. 21.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
24. 24.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
29. 29.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
30. 30.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 33 (d) 17
31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 14 (d) 48
32. 32.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 4 (c) 106 (d) 16
33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 289 (d) 145
34. 34.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
35. 35.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 26 (d) 12
36. 36.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
37. 37.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
38. 38.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 3 and 9 (d) 1 and 11
39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7056 (c) 6416 (d) 7066
40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4

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