Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 872 · 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 = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 50 (d) 6
2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 784 (d) 896
3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
4. 4.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 2035 (d) 1625
5. 5.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
6. 6.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 4 (d) 2
7. 7.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 9 (d) 8
8. 8.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 4925 (d) 5635
9. 9.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 2 and 7 (c) 1 and 14 (d) 1 and 8
10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
12. 12.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6724 (d) 6084
13. 13.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
14. 14.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
15. 15.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
16. 16.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 1225 (d) 875
17. 17.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
18. 18.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 5 (d) 73
19. 19.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 2 (d) 100
20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
21. 21.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 223 (d) 289
22. 22.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
24. 24.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)
25. 25.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)
26. 26.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 91 (c) 121 (d) 181
27. 27.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 64 (d) 63
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
29. 29.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4499 (d) 4489
30. 30.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 11025 (d) 9025
31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
32. 32.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
33. 33.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 16
34. 34.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 4 (c) 80 (d) 144
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
37. 37.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256
38. 38.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
39. 39.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
40. 40.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598

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