Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 418 · 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 + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1764 (c) 1774 (d) 1604
3. 3.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 127 (d) 253
4. 4.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 146 (c) 366 (d) 201
5. 5.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 5329 (d) 4499
6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
7. 7.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 35 (d) 13
8. 8.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
10. 10.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 90 (d) 198
11. 11.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
13. 13.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 19 (d) 37
14. 14.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 121 (d) 61
15. 15.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)
16. 16.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 53 (c) 15 (d) 55
17. 17.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
19. 19.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
20. 20.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
21. 21.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)
22. 22.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4096 (d) 4106
23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
24. 24.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 1225 (d) 2025
25. 25.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 10 (d) 4
26. 26.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 9025 (d) 11025
28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
29. 29.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 4 (d) 2
30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1225 (c) 2025 (d) 1235
31. 31.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 256 (d) 208
32. 32.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
33. 33.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
34. 34.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 31 (d) 12
35. 35.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3481 (d) 3601
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
37. 37.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
38. 38.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 1 and 36 (d) 4 and 9
39. 39.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 3 and 21 (d) 1 and 63
40. 40.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041

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