Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 253 · 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 + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
2. 2.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 2 (d) 100
3. 3.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 361 (d) 537
4. 4.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2209 (d) 2219
5. 5.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
6. 6.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 217 (d) 1
7. 7.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 14 (d) 45
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
10. 10.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 153 (d) 189
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
14. 14.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
15. 15.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 851 (d) 961
16. 16.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 2 and 6 (d) 1 and 12
17. 17.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 361 (d) 185
18. 18.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
19. 19.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
21. 21.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
22. 22.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 177 (d) 65
23. 23.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
24. 24.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7754 (d) 7744
25. 25.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 64 (d) 8
26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
27. 27.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 5625 (d) 4225
28. 28.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 14 (c) 8 (d) 16
29. 29.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 249 (c) 129 (d) 169
30. 30.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
31. 31.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 1 (d) 41
32. 32.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1936 (d) 1616
33. 33.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7579 (d) 7569
34. 34.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 8 (d) 12
35. 35.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
36. 36.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
38. 38.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
39. 39.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101

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