Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 244 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
3. 3.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169
4. 4.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
5. 5.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
7. 7.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
8. 8.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 7569 (d) 8091
9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
10. 10.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 1225 (d) 635
11. 11.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1225 (d) 1575
12. 12.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 2035 (d) 1625
13. 13.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
14. 14.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
15. 15.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
17. 17.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 180 (d) 6
18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
19. 19.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)
20. 20.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4899 (d) 4761
21. 21.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 384 (d) 256
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 7744 (d) 8464
23. 23.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 36 (d) 35
24. 24.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 361 (d) 273
26. 26.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)
27. 27.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 225 (d) 25
28. 28.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 157 (d) 145
29. 29.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6401 (d) 6571
31. 31.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
32. 32.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5051 (d) 5041
33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
34. 34.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
36. 36.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
37. 37.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 1 and 13 (d) 5 and 9
38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
39. 39.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
40. 40.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 2 and 4 (d) 1 and 8

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