Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 464 · 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 − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
3. 3.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
4. 4.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
7. 7.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
8. 8.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175
9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 6 (d) 8
10. 10.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
11. 11.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 208 (d) 16
12. 12.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 2035 (d) 1625
13. 13.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 144 (d) 74
14. 14.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
15. 15.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
16. 16.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 1 and 4 (d) 2 and 4
17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
18. 18.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
20. 20.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 34 (c) 4 (d) 64
21. 21.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 4096 (d) 3146
22. 22.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 64 (c) 196 (d) 8
23. 23.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
24. 24.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)
25. 25.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)
26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
27. 27.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
28. 28.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
29. 29.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 10 (d) 24
30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
31. 31.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 7931 (d) 8099
32. 32.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 65 (d) 9
33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 225 (d) 5
34. 34.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 5 (d) 9
35. 35.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 9
36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
39. 39.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
40. 40.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 35 (d) 36

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