Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 677 · 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 + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
3. 3.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
5. 5.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 12 (d) 8
6. 6.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
7. 7.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
8. 8.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6094 (d) 6396
9. 9.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 3 and 4 (c) 1 and 10 (d) 1 and 6
10. 10.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 3 and 4 (d) 1 and 12
11. 11.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 32 (d) 11
12. 12.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
13. 13.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 117 (c) 171 (d) 333
14. 14.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
15. 15.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 65 (c) 7 (d) 81
16. 16.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
17. 17.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 916 (d) 676
18. 18.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
19. 19.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 5476 (c) 4356 (d) 4884
20. 20.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 6 (d) 7
21. 21.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)
22. 22.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)
23. 23.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
24. 24.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
25. 25.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 116 (d) 156
26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
27. 27.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
28. 28.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)
29. 29.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 4 (d) 2
30. 30.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
32. 32.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 3 and 6 (d) 1 and 18
33. 33.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 8 (c) 196 (d) 64
34. 34.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
35. 35.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
36. 36.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 15 (d) 14
37. 37.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 256 (d) 208
38. 38.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 3 (d) 9
39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
40. 40.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17

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