Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 805 · 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.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
2. 2.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
5. 5.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 54 (d) 55
6. 6.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
7. 7.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
8. 8.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 841 (d) 901
9. 9.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 116 (d) 156
10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
11. 11.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
13. 13.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
14. 14.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4489 (d) 4909
15. 15.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
16. 16.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 149 (c) 9 (d) 3
17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
19. 19.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 41 (d) 42
20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
21. 21.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 43 (c) 61 (d) 49
22. 22.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 196 (c) 292 (d) 148
23. 23.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 657 (d) 333
24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
26. 26.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
27. 27.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
28. 28.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 3 and 9 (c) 1 and 27 (d) 2 and 10
29. 29.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 1 and 18 (d) 3 and 6
30. 30.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 81 (d) 9
31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
32. 32.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
33. 33.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3721 (d) 3481
34. 34.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
35. 35.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 147 (c) 125 (d) 213
36. 36.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
37. 37.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 235 (d) 225
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 9
40. 40.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)

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