Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 507 · 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 21²? (a) 451 (b) 441 (c) 401 (d) 361
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
3. 3.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
5. 5.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 81 (d) 9
6. 6.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
7. 7.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 9 (d) 7
8. 8.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1521 (d) 1681
9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
10. 10.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3721 (d) 3731
11. 11.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1764 (d) 1454
12. 12.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 169 (d) 229
13. 13.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
14. 14.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
15. 15.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 34 (d) 4
16. 16.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)
17. 17.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 73 (c) 97 (d) 169
18. 18.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 144 (d) 124
19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
21. 21.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2491 (d) 2809
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
23. 23.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
24. 24.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10009 (d) 10609
25. 25.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 266 (d) 256
26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
29. 29.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 6384 (d) 5786
30. 30.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 292 (d) 196
31. 31.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3616 (c) 3136 (d) 4106
32. 32.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 180 (d) 36
33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
34. 34.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 37 (c) 25 (d) 49
35. 35.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 1 and 12 (d) 5 and 8
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
38. 38.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
39. 39.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 2 and 4 (c) 1 and 4 (d) 1 and 6
40. 40.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)

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