Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 598 · 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.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
2. 2.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
4. 4.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
7. 7.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 25 (d) 289
8. 8.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 14 (d) 42
9. 9.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
10. 10.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 196 (d) 100
11. 11.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 81 (d) 45
12. 12.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 2 and 10 (d) 4 and 5
13. 13.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6724 (d) 6396
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
15. 15.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3249 (d) 3969
16. 16.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 12 (c) 11 (d) 27
17. 17.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
18. 18.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 121 (d) 91
19. 19.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
21. 21.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)
22. 22.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
23. 23.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
25. 25.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
26. 26.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
27. 27.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
28. 28.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
29. 29.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521
30. 30.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 1 and 54 (d) 2 and 27
31. 31.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 8 (d) 12
32. 32.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 11035 (d) 9025
33. 33.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 2 and 12 (d) 1 and 24
34. 34.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
35. 35.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
37. 37.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)
38. 38.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
39. 39.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 4 (d) 16
40. 40.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12

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