Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 461 · 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 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
2. 2.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 57 (d) 15
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
8. 8.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 9035 (d) 7225
9. 9.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 25 (c) 5 (d) 125
10. 10.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 34 (d) 64
11. 11.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 91 (d) 121
12. 12.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 225 (d) 425
13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 5625 (d) 4225
14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 9025 (d) 11025
15. 15.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
16. 16.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 8 (d) 64
17. 17.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5329 (c) 5339 (d) 4489
18. 18.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 5 and 6 (d) 1 and 30
19. 19.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
20. 20.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 8 (d) 7
21. 21.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 361 (d) 401
25. 25.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 14 (d) 12
26. 26.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 13 (c) 1 and 45 (d) 5 and 9
27. 27.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 9975 (d) 11025
28. 28.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)
29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
30. 30.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 13 (d) 19
31. 31.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
32. 32.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
33. 33.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
35. 35.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
36. 36.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3146 (d) 3584
37. 37.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 8 (c) 82 (d) 100
38. 38.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 324 (d) 180
39. 39.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 25 (d) 19
40. 40.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404

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