Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 229 · 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 = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
2. 2.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
3. 3.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)
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
5. 5.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 180 (d) 468
6. 6.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)
7. 7.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 4225 (c) 3035 (d) 3575
8. 8.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 2 and 6 (d) 1 and 15
9. 9.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
10. 10.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 7 (d) 8
11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
12. 12.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
13. 13.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
14. 14.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 81 (c) 9 (d) 169
15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 256 (d) 384
17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
18. 18.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 6 (d) 8
19. 19.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
20. 20.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
21. 21.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
22. 22.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
23. 23.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 13 (c) 15 (d) 9
24. 24.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8104 (d) 8464
25. 25.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 8 (c) 82 (d) 100
26. 26.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 9999 (d) 10201
27. 27.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)
28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
29. 29.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
31. 31.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
34. 34.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 36 (d) 26
35. 35.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 19 (d) 17
36. 36.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
37. 37.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
39. 39.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 8281 (d) 7921
40. 40.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)

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