Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 24 · 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 is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 27 (d) 11
2. 2.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 49 (d) 73
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
4. 4.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 1 and 48 (d) 6 and 8
5. 5.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 166 (d) 144
6. 6.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10609 (d) 10009
7. 7.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
8. 8.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
11. 11.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
12. 12.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
13. 13.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 1225 (d) 625
14. 14.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 169 (d) 337
15. 15.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
16. 16.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 739 (d) 729
17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
18. 18.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 875 (d) 625
19. 19.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 7
20. 20.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
22. 22.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)
23. 23.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
24. 24.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
25. 25.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)
26. 26.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
29. 29.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 235 (d) 375
30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
31. 31.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 36 (d) 35
32. 32.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041
33. 33.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
34. 34.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
37. 37.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
38. 38.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 121 (d) 61
40. 40.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7

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