Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 437 · 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 = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
2. 2.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 244 (c) 172 (d) 196
3. 3.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)
4. 4.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
5. 5.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
6. 6.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 851 (d) 899
7. 7.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
8. 8.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
10. 10.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
11. 11.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 5 (d) 169
12. 12.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 1 and 27 (c) 2 and 10 (d) 3 and 9
13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
14. 14.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 9 (d) 89
15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
16. 16.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
18. 18.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)
19. 19.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 25 (d) 19
20. 20.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 14 (d) 44
21. 21.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 144 (d) 104
22. 22.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
23. 23.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 193 (c) 382 (d) 256
24. 24.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 7 (d) 8
25. 25.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
27. 27.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 137 (d) 313
28. 28.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)
29. 29.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1859 (d) 1369
30. 30.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
34. 34.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 16 (d) 15
35. 35.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
36. 36.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
37. 37.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
39. 39.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
40. 40.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 22 (d) 10

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