Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 407 · 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 + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
3. 3.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
5. 5.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 1625 (d) 2035
6. 6.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 8 (d) 14
7. 7.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
8. 8.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 3035 (d) 2525
9. 9.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
10. 10.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 149 (d) 289
11. 11.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
12. 12.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 36 (d) 26
13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 400 (d) 202
14. 14.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 729 (d) 909
15. 15.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 256 (c) 160 (d) 64
16. 16.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
19. 19.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 18 (d) 17
20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
21. 21.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
22. 22.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1225 (c) 2025 (d) 1235
23. 23.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 5 and 7 (d) 1 and 35
24. 24.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
25. 25.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)
26. 26.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)
27. 27.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
28. 28.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
31. 31.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 10 (d) 196
32. 32.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 2 and 20 (d) 1 and 12
33. 33.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
34. 34.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 425 (d) 225
35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
36. 36.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 376 (d) 196
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.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 1 and 8 (d) 2 and 9
40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181

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