Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 571 · 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.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
2. 2.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1296 (d) 1936
3. 3.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 28 (d) 26
4. 4.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 297 (d) 153
5. 5.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 19 (c) 1 (d) 13
6. 6.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 198 (c) 117 (d) 90
7. 7.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 2 (d) 4
8. 8.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 64 (d) 100
9. 9.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3969 (d) 3979
10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
11. 11.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 64 (d) 50
12. 12.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
13. 13.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 49 (d) 14
14. 14.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
15. 15.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)
16. 16.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
17. 17.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 100 (c) 76 (d) 148
18. 18.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
19. 19.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
20. 20.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 1225 (d) 2035
21. 21.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 23 (c) 25 (d) 10
22. 22.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1296 (d) 1936
23. 23.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
25. 25.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
26. 26.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
27. 27.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
28. 28.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 145 (c) 193 (d) 169
30. 30.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
32. 32.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 7235 (d) 5625
33. 33.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 244 (d) 172
34. 34.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 7921 (d) 8291
35. 35.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
37. 37.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 1 and 12 (d) 2 and 21
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
39. 39.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3491 (d) 3599
40. 40.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12

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