Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 169 · 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 29²? (a) 841 (b) 961 (c) 899 (d) 851
2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
3. 3.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 22 (d) 10
4. 4.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 4096 (d) 3136
5. 5.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)
6. 6.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
7. 7.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 7066 (d) 6416
8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
9. 9.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
11. 11.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
12. 12.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 3 and 15 (d) 1 and 13
13. 13.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 4 (d) 324
14. 14.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
15. 15.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
16. 16.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 25 (d) 3
17. 17.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
20. 20.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6084 (c) 6724 (d) 6094
21. 21.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
22. 22.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
23. 23.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
24. 24.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 7 (d) 5
25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 256 (d) 576
26. 26.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
27. 27.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 9 (d) 11
28. 28.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 9984 (d) 10816
29. 29.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
30. 30.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9035 (d) 9025
32. 32.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
34. 34.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
35. 35.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
36. 36.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
37. 37.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
38. 38.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
39. 39.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
40. 40.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277

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