Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 201 · 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.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 2525 (d) 3025
2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
4. 4.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
5. 5.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 136 (d) 4
6. 6.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
7. 7.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
8. 8.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 81 (d) 1
9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
10. 10.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 6 and 7 (d) 1 and 12
11. 11.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
12. 12.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 172 (c) 244 (d) 196
13. 13.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
14. 14.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 169 (d) 125
15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
16. 16.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
17. 17.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2509 (d) 2819
18. 18.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 15 (c) 9 (d) 14
19. 19.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
20. 20.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3249 (d) 3969
21. 21.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 337 (d) 169
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
23. 23.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 7235 (d) 6425
24. 24.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 7 (d) 49
25. 25.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 9 (d) 81
26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
27. 27.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 7 (d) 6
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
30. 30.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1454 (d) 1444
31. 31.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 2 and 15 (d) 1 and 30
32. 32.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
34. 34.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)
35. 35.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 64 (d) 82
36. 36.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 253 (d) 169
38. 38.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 6 (d) 5
39. 39.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 49 (c) 225 (d) 7
40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 404 (d) 324

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