Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 118 · 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 − 10x + 21 = 0? (a) 20 (b) 10 (c) 21 (d) 22
2. 2.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 124 (d) 144
3. 3.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
4. 4.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2516 (d) 2926
5. 5.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
7. 7.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)
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
9. 9.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 11 (d) 13
10. 10.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 5 (c) 169 (d) 97
11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
13. 13.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 3 and 6 (d) 1 and 18
14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
15. 15.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3259 (d) 3591
16. 16.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
18. 18.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 137 (c) 313 (d) 225
19. 19.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
21. 21.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
22. 22.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 73 (d) 169
23. 23.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 1 (d) 145
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
26. 26.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
27. 27.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 25 (d) 5
28. 28.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
29. 29.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 9025 (d) 11025
30. 30.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
31. 31.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
32. 32.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
33. 33.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 52 (c) 148 (d) 100
34. 34.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 635 (d) 225
35. 35.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 361 (d) 185
36. 36.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
37. 37.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
38. 38.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 5 (d) 8
39. 39.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
40. 40.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 9 (d) 3

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