Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 236 · 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 + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
2. 2.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
3. 3.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 20 (d) 8
4. 4.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
6. 6.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
7. 7.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2219 (c) 2491 (d) 2809
8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
9. 9.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
10. 10.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 196 (d) 276
11. 11.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)
12. 12.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 841 (d) 899
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
14. 14.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
16. 16.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 244 (d) 484
18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 841 (d) 971
19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 8 (d) 7
20. 20.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 64 (d) 34
21. 21.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 8 (c) 160 (d) 256
22. 22.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
23. 23.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 11 (d) 28
24. 24.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
25. 25.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
26. 26.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
27. 27.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 196
28. 28.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
29. 29.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 8 (d) 3 and 6
30. 30.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 19 (c) 11 (d) 18
31. 31.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
32. 32.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 1 (d) 145
33. 33.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 7 (d) 49
34. 34.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
35. 35.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 7 (d) 5
36. 36.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1225 (d) 1625
37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
40. 40.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262

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