Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 527 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
2. 2.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
4. 4.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
8. 8.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 16 (c) 100 (d) 58
9. 9.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 64 (c) 196 (d) 130
10. 10.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
11. 11.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 657 (d) 333
12. 12.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 468 (d) 252
13. 13.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4489 (d) 4909
14. 14.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 592 (d) 400
16. 16.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 30 (d) 1 and 10
17. 17.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 7 (d) 8
18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
19. 19.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 141 (d) 121
20. 20.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 244 (d) 148
21. 21.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8836 (c) 8846 (d) 7396
22. 22.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 26 (d) 36
23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
24. 24.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
25. 25.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
26. 26.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 9 (d) 185
27. 27.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
28. 28.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
29. 29.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
30. 30.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 9 (d) 21
31. 31.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 2 and 16 (d) 1 and 32
32. 32.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
33. 33.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
34. 34.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)
35. 35.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 42 (d) 12
38. 38.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 35 (d) 12
39. 39.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
40. 40.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399

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