Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 756 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
2. 2.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 5 (d) 1 and 15
3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
4. 4.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
5. 5.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
7. 7.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 262 (d) 163
8. 8.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2025 (c) 2035 (d) 3025
9. 9.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3481 (d) 3601
10. 10.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 17 (d) 25
11. 11.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 208 (d) 160
12. 12.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
13. 13.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
17. 17.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
18. 18.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 3 and 4 (d) 1 and 6
19. 19.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 875 (d) 1225
20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
21. 21.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)
22. 22.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 8075 (d) 9025
23. 23.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
25. 25.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 2 and 8 (d) 1 and 9
26. 26.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 6 (d) 7
27. 27.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
28. 28.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
29. 29.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 899 (d) 961
30. 30.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 5625 (d) 7225
31. 31.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
32. 32.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 82 (c) 8 (d) 100
33. 33.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2916 (c) 2116 (d) 2516
34. 34.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
35. 35.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 169 (d) 113
36. 36.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 289 (d) 149
37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
38. 38.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3844 (d) 3596
39. 39.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 16 (d) 4
40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18

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