Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 254 · 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 − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
2. 2.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
3. 3.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
4. 4.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 170 (d) 478
5. 5.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
6. 6.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 14 (d) 12
7. 7.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 9801 (c) 10211 (d) 10001
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
10. 10.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 97 (d) 169
11. 11.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
12. 12.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2219 (d) 2209
13. 13.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 2 and 9 (d) 3 and 6
14. 14.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
17. 17.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
18. 18.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
19. 19.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 7 (d) 49
20. 20.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 49 (c) 37 (d) 25
21. 21.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 4225 (d) 3035
22. 22.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 39 (c) 13 (d) 41
27. 27.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2809 (d) 2509
28. 28.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
29. 29.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 729 (d) 909
30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 289 (d) 529
31. 31.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 484 (d) 164
32. 32.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 45 (c) 1 and 13 (d) 3 and 15
33. 33.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 409 (d) 229
34. 34.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
35. 35.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
36. 36.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8101 (d) 8281
37. 37.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3731 (d) 3721
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
39. 39.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
40. 40.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 7 (c) 9 (d) 5

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