Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 46 · 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 + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
4. 4.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 5 (d) 7
5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
7. 7.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 15 (d) 14
9. 9.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
10. 10.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 1 and 12 (d) 2 and 20
11. 11.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 208 (c) 4 (d) 400
12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
13. 13.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
14. 14.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 4 (d) 324
15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
16. 16.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1591 (d) 1849
17. 17.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
18. 18.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 361 (d) 3
19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
20. 20.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10001 (d) 10201
21. 21.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
22. 22.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 106 (d) 151
23. 23.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
24. 24.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 117 (d) 3
25. 25.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
26. 26.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 10 (d) 23
27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
28. 28.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 3 and 8 (c) 1 and 10 (d) 2 and 12
29. 29.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 5 (d) 25
30. 30.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 80 (c) 144 (d) 4
31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
32. 32.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
33. 33.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
34. 34.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 100 (c) 58 (d) 4
35. 35.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 529 (d) 193
36. 36.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4925 (d) 4225
37. 37.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 15 (c) 1 and 63 (d) 3 and 21
38. 38.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
39. 39.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
40. 40.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 851 (d) 961

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