Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 153 · 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.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 12 (c) 14 (d) 13
2. 2.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
3. 3.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)
4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
5. 5.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
6. 6.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
7. 7.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
8. 8.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1616 (c) 1296 (d) 1946
9. 9.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 144 (c) 64 (d) 8
10. 10.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
11. 11.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
12. 12.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
13. 13.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
14. 14.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 6425 (d) 5625
15. 15.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)
16. 16.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 916 (d) 1166
17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10001 (d) 10211
18. 18.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
19. 19.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 784 (d) 1024
20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
21. 21.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
22. 22.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 10609 (c) 9991 (d) 9419
23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175
24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
25. 25.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2704 (d) 2496
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
27. 27.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 29 (c) 49 (d) 3
28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
30. 30.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6899 (d) 6409
31. 31.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 26 (d) 12
32. 32.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6391 (d) 6889
33. 33.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 7 (c) 8 (d) 10
34. 34.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)
35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
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.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)
39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
40. 40.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256

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