Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 446 · 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 are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 54 (d) 1 and 14
2. 2.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 8 (d) 3 and 6
3. 3.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
5. 5.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3249 (c) 3609 (d) 3979
6. 6.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 25 (d) 169
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 10 (d) 4
8. 8.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
9. 9.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)
10. 10.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
11. 11.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
12. 12.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 31 (d) 36
13. 13.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 109 (d) 214
14. 14.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 15 (d) 16
15. 15.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 8 (d) 1 and 5
16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
17. 17.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
18. 18.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025
19. 19.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 529 (d) 289
20. 20.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 68 (d) 36
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
22. 22.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
24. 24.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 145 (d) 289
25. 25.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 125 (d) 25
26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
28. 28.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
29. 29.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
32. 32.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 64 (c) 256 (d) 160
33. 33.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
34. 34.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1859 (c) 1369 (d) 1609
35. 35.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
36. 36.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 41 (d) 61
37. 37.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
38. 38.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
39. 39.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 399 (d) 361
40. 40.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 875 (d) 635

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