Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 213 · 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 is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 12 (d) 10
2. 2.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)
3. 3.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
4. 4.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 144 (d) 109
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
6. 6.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
7. 7.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10004 (d) 10414
8. 8.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 40 (c) 14 (d) 13
9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
12. 12.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
13. 13.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
14. 14.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 4904 (d) 5184
15. 15.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 45 (d) 1 and 13
17. 17.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
18. 18.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 7 (d) 1 and 12
19. 19.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
20. 20.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 21 (d) 19
21. 21.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 3 and 4 (d) 1 and 12
22. 22.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
24. 24.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 97 (d) 241
25. 25.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 121 (d) 81
26. 26.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 25 (d) 193
27. 27.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
28. 28.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 1 and 18 (d) 2 and 9
29. 29.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 153 (d) 225
30. 30.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 1 and 8 (d) 4 and 5
31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
32. 32.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 5 and 6 (d) 2 and 15
33. 33.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262
34. 34.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 3 and 7 (d) 1 and 21
35. 35.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
37. 37.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
38. 38.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 117 (d) 333
39. 39.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)

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