Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 409 · 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 are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
2. 2.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
3. 3.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
4. 4.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
5. 5.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
6. 6.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
8. 8.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
9. 9.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
10. 10.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 64 (d) 50
11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
12. 12.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 37 (d) 61
13. 13.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
14. 14.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 125 (d) 325
15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
16. 16.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
17. 17.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 169 (c) 147 (d) 125
18. 18.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
20. 20.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 1 and 4 (d) 2 and 3
21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
22. 22.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 82 (c) 100 (d) 8
23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
24. 24.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 409 (c) 289 (d) 169
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 7 (d) 5
27. 27.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
28. 28.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 16 (d) 4
29. 29.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
31. 31.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8281 (d) 8099
32. 32.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
33. 33.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 4884 (d) 5476
34. 34.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
35. 35.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 12 (d) 28
36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
38. 38.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)
39. 39.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18
40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16

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