Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 165 · 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.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
2. 2.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 361 (d) 451
3. 3.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
4. 4.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3591 (c) 3259 (d) 3969
5. 5.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
6. 6.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 148 (d) 100
7. 7.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 11 (d) 1 and 32
8. 8.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
9. 9.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 7 (c) 8 (d) 6
10. 10.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
11. 11.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
12. 12.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 169 (c) 147 (d) 125
13. 13.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 5 (c) 25 (d) 225
14. 14.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 1 and 21 (c) 3 and 7 (d) 2 and 8
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
16. 16.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
17. 17.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 7744 (d) 8096
18. 18.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 25 (d) 3
19. 19.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10009 (d) 10609
20. 20.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
21. 21.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5339 (d) 5329
22. 22.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 25 (c) 33 (d) 17
23. 23.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)
24. 24.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)
25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
26. 26.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
29. 29.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
30. 30.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
32. 32.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 289 (d) 217
34. 34.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 122 (d) 144
35. 35.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 160 (d) 352
36. 36.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 11 (d) 10
37. 37.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 4 (d) 6
38. 38.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2401 (d) 2411
39. 39.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 24 (c) 1 and 10 (d) 2 and 12
40. 40.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 93 (d) 121

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad