Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 366 · 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 = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 144 (c) 16 (d) 80
2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
4. 4.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
5. 5.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 8 (d) 20
6. 6.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
8. 8.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 35 (d) 1 and 11
9. 9.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 25 (d) 5
10. 10.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
11. 11.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
12. 12.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 37 (d) 25
13. 13.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 10609 (d) 9419
14. 14.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 29 (d) 69
15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 289 (d) 539
16. 16.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 14 (d) 15
17. 17.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
18. 18.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
19. 19.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
20. 20.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 361 (d) 3
21. 21.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
23. 23.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 4884 (d) 5476
24. 24.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 84 (d) 132
25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
26. 26.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
28. 28.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 225 (d) 625
29. 29.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 2 (d) 164
30. 30.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1584 (d) 1936
31. 31.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
32. 32.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 15 (d) 63
33. 33.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
34. 34.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 11 (d) 24
35. 35.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
36. 36.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
39. 39.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
40. 40.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15

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