Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 333 · 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 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
2. 2.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
3. 3.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 2 and 16 (d) 4 and 8
4. 4.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
5. 5.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 11 (d) 12
6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
7. 7.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 6409 (d) 5929
8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2704 (d) 2304
9. 9.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 148 (d) 172
10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
11. 11.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
12. 12.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
15. 15.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
19. 19.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4899 (d) 4771
20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
21. 21.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 4 and 6 (d) 2 and 12
22. 22.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 2 and 21 (d) 1 and 42
23. 23.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
24. 24.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 3 and 4 (d) 1 and 6
25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 61 (d) 81
26. 26.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 48 (d) 13
27. 27.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
28. 28.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
29. 29.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 529 (d) 391
30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
31. 31.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
32. 32.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3481 (d) 3599
33. 33.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 289 (d) 217
35. 35.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)
36. 36.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
39. 39.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 289
40. 40.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202

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