Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 745 · 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 = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 6 (d) 50
2. 2.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 8
3. 3.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 73 (d) 97
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
5. 5.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
6. 6.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1859 (d) 1849
7. 7.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1599 (d) 1521
8. 8.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 16 (d) 15
9. 9.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)
10. 10.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2926 (d) 2916
11. 11.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
12. 12.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
14. 14.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
15. 15.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
17. 17.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
18. 18.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
21. 21.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
24. 24.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 25 (d) 24
25. 25.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 4225 (d) 5625
26. 26.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 13 (d) 1 and 48
27. 27.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
28. 28.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
29. 29.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 5 (c) 13 (d) 7
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
31. 31.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 5 (c) 37 (d) 49
32. 32.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
33. 33.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
34. 34.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
35. 35.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 144 (d) 184
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 361 (d) 401
38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
39. 39.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 37 (d) 25
40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)

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