Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 508 · 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 = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
6. 6.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
7. 7.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
8. 8.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)
9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
10. 10.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 12 (d) 3 and 4
11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
12. 12.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 4 (d) 64
13. 13.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1609 (d) 1859
14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
16. 16.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
17. 17.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
18. 18.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 196 (c) 286 (d) 151
19. 19.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
21. 21.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
22. 22.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041
23. 23.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
25. 25.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 529 (d) 361
26. 26.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9035 (c) 7225 (d) 9025
27. 27.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)
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7921 (d) 7931
30. 30.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
31. 31.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 40 (d) 14
32. 32.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 164 (d) 2
33. 33.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
34. 34.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
35. 35.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
36. 36.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
38. 38.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 249 (d) 89
39. 39.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
40. 40.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 28 (d) 20

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