Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 242 · 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 = 18 and ab = 72, what is a² + b²? (a) 180 (b) 252 (c) 468 (d) 324
2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4916 (d) 4356
3. 3.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 145 (c) 121 (d) 11
4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
5. 5.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)
6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
7. 7.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 18 (d) 9
8. 8.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)
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
10. 10.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10016 (d) 10816
11. 11.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
12. 12.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
13. 13.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
15. 15.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 50 (d) 64
16. 16.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
17. 17.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
18. 18.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3969 (c) 3591 (d) 3249
19. 19.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
20. 20.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
22. 22.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 89 (d) 249
23. 23.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 4 and 9 (d) 2 and 18
24. 24.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
26. 26.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
28. 28.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 1625 (d) 2025
29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
30. 30.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 5 (d) 25
31. 31.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7569 (c) 8649 (d) 7579
32. 32.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 899 (d) 961
33. 33.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
34. 34.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
35. 35.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
38. 38.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 37 (d) 35
39. 39.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 16 (c) 15 (d) 10
40. 40.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3364 (c) 3844 (d) 3374

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