Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 298 · 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) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
2. 2.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
3. 3.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
4. 4.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
5. 5.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
6. 6.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 5486 (d) 4916
7. 7.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
8. 8.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
9. 9.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 63 (d) 45
10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1454 (d) 1596
11. 11.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
12. 12.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 2 and 14 (d) 4 and 7
13. 13.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)
14. 14.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 196 (d) 4
15. 15.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)
16. 16.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 180 (d) 324
17. 17.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 10 (d) 28
18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
20. 20.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 4 and 6 (c) 2 and 12 (d) 1 and 9
21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
22. 22.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
24. 24.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
25. 25.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
27. 27.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
28. 28.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3364 (d) 3374
29. 29.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 6 (d) 10
30. 30.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 3 and 5 (d) 1 and 7
31. 31.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 20 (c) 36 (d) 2
32. 32.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
33. 33.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251
34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 289 (c) 217 (d) 145
35. 35.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6561 (d) 6241
36. 36.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
37. 37.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 7 (d) 9
38. 38.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)
39. 39.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2126 (d) 2484
40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)

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