Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 206 · 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.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 181 (d) 361
2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
3. 3.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
4. 4.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 7 (d) 5
5. 5.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
6. 6.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
7. 7.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)
8. 8.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
10. 10.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 7 (d) 6
11. 11.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
12. 12.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 635 (d) 425
13. 13.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
14. 14.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 324 (c) 468 (d) 180
15. 15.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
16. 16.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 22 (d) 20
17. 17.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 2 and 10 (d) 1 and 11
18. 18.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
19. 19.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
21. 21.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
22. 22.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6571 (d) 6401
23. 23.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2916 (d) 2926
24. 24.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 101 (d) 81
25. 25.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 4771 (d) 5041
26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
27. 27.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 1225 (d) 875
28. 28.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 13 (d) 7
29. 29.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 82 (d) 64
30. 30.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2209 (d) 2491
31. 31.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
32. 32.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 11 (d) 32
33. 33.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 7 (d) 11
34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 12 (d) 42
36. 36.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 169 (d) 7
37. 37.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)
38. 38.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 1225 (d) 925
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
40. 40.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 2 and 14 (c) 1 and 28 (d) 4 and 7

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