Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 713 · 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.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 180 (c) 324 (d) 468
2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
4. 4.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
5. 5.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)
6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
7. 7.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
8. 8.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
9. 9.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 144 (d) 109
11. 11.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 289 (d) 157
12. 12.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 361 (d) 1
13. 13.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 100 (c) 79 (d) 142
14. 14.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 91 (d) 1
15. 15.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9
16. 16.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 9 (c) 3 (d) 149
17. 17.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 49 (d) 85
18. 18.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1584 (d) 1296
19. 19.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 4 (d) 16
20. 20.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
22. 22.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 65 (c) 81 (d) 7
23. 23.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2209 (d) 2509
24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 361 (d) 277
27. 27.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 36 (d) 180
28. 28.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 2 and 10 (d) 1 and 27
29. 29.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
30. 30.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 10 (d) 16
31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
32. 32.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 12 (d) 8
33. 33.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 147 (d) 213
34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
35. 35.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3969 (d) 3249
36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 6 (c) 144 (d) 36
37. 37.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 9604 (d) 10404
38. 38.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1454 (d) 1596
39. 39.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 5 (d) 1
40. 40.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35

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