Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 9 · 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 + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
2. 2.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
4. 4.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8091 (d) 8649
5. 5.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 289 (c) 529 (d) 391
6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
7. 7.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 137 (c) 225 (d) 313
8. 8.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
9. 9.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 64 (c) 36 (d) 50
10. 10.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
11. 11.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 784 (c) 794 (d) 1024
12. 12.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 10 (c) 11 (d) 12
13. 13.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 90 (d) 117
14. 14.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)
15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
17. 17.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 21
18. 18.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
19. 19.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
21. 21.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 68 (c) 84 (d) 100
22. 22.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 3616 (d) 4096
23. 23.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
24. 24.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181
26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
27. 27.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
28. 28.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
29. 29.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
30. 30.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
31. 31.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 2 and 5 (c) 1 and 10 (d) 3 and 4
32. 32.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 10609 (d) 9419
33. 33.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 25 (d) 17
34. 34.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 100 (c) 8 (d) 82
35. 35.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 625 (d) 635
36. 36.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
38. 38.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9614 (c) 9604 (d) 9996
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)

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