Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 16 · 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.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
2. 2.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 4 (d) 100
3. 3.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 1 and 54 (d) 6 and 9
4. 4.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
5. 5.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
6. 6.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 1 and 10 (d) 2 and 15
7. 7.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 225 (d) 169
8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
10. 10.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 153 (d) 189
11. 11.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 289
12. 12.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 13 (d) 22
13. 13.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
14. 14.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 9 (d) 1 and 21
15. 15.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
16. 16.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
18. 18.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 13 (d) 14
19. 19.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 4 (d) 34
20. 20.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)
21. 21.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 54 (d) 16
22. 22.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
23. 23.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
24. 24.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)
25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3364 (d) 3374
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
27. 27.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
28. 28.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
29. 29.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
30. 30.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 11
31. 31.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
32. 32.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
33. 33.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
34. 34.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
35. 35.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 157 (c) 193 (d) 169
36. 36.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9614 (d) 9604
37. 37.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 8104 (d) 7744
38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
39. 39.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 36 (d) 26
40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)

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