Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 266 · 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 are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 13 (c) 1 and 48 (d) 2 and 24
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
4. 4.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 97 (c) 169 (d) 241
5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
6. 6.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 223 (d) 157
7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
8. 8.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 28 (c) 29 (d) 11
9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
10. 10.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
11. 11.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4761 (d) 4901
12. 12.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10404 (d) 10414
13. 13.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
14. 14.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
15. 15.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)
16. 16.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 164 (d) 324
17. 17.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 4 (d) 16
18. 18.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
19. 19.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 144 (d) 74
20. 20.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 144 (c) 10 (d) 122
21. 21.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 729 (d) 909
22. 22.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 14 (d) 13
23. 23.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 10 (d) 11
24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
25. 25.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10816 (d) 10826
26. 26.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 1 and 40 (d) 2 and 20
27. 27.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
30. 30.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2209 (d) 2509
31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9614 (d) 9604
32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
33. 33.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
34. 34.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
36. 36.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 49 (d) 25
37. 37.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
40. 40.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 11 (c) 21 (d) 10

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