Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 646 · 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 + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
3. 3.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
4. 4.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)
5. 5.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
6. 6.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10619 (c) 9409 (d) 10009
7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 2 (d) 202
8. 8.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 10 (d) 6
9. 9.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 289 (d) 7
10. 10.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 101 (d) 141
11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
12. 12.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 3 and 5 (d) 1 and 15
13. 13.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 121 (d) 73
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
15. 15.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
16. 16.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
17. 17.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
18. 18.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)
19. 19.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
20. 20.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2516 (d) 2926
21. 21.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 5 and 6 (c) 1 and 30 (d) 2 and 15
22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
23. 23.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
24. 24.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 9 (d) 11
25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
26. 26.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
27. 27.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
29. 29.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
30. 30.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
34. 34.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
35. 35.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
37. 37.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 4 (c) 196 (d) 16
38. 38.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
39. 39.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 592 (d) 208
40. 40.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441

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