Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 666 · 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 = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
2. 2.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 10609 (c) 9409 (d) 9991
3. 3.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2499 (d) 2601
4. 4.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 13 (d) 15
5. 5.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 25 (d) 1
6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
7. 7.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 289 (d) 229
8. 8.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 23 (c) 10 (d) 25
9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
12. 12.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 17 (c) 21 (d) 25
13. 13.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
15. 15.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
18. 18.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 56 (d) 15
19. 19.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 81 (d) 53
20. 20.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 8 (d) 9
21. 21.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)
22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
23. 23.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 4 (d) 36
24. 24.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 29 (c) 11 (d) 27
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
26. 26.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 184 (d) 144
27. 27.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 64 (c) 196 (d) 8
28. 28.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 324 (d) 164
29. 29.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 11 (c) 24 (d) 25
30. 30.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
31. 31.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 100 (d) 148
32. 32.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454
33. 33.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 598 (d) 400
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 909 (d) 1099
35. 35.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
36. 36.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
37. 37.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
38. 38.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
39. 39.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 49 (d) 94
40. 40.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 20 (c) 36 (d) 4

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