Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 765 · 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 − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
2. 2.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 9811 (d) 10201
3. 3.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 8 (d) 6
4. 4.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
5. 5.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 148 (d) 52
6. 6.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
7. 7.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6251 (d) 6561
8. 8.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 4235 (d) 3025
9. 9.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 3616 (d) 4096
10. 10.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)
11. 11.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 117 (d) 171
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
13. 13.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 20 (d) 8
14. 14.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
16. 16.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 193 (d) 361
17. 17.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 9 (d) 11
18. 18.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 4899 (d) 5041
19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
20. 20.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
22. 22.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 35 (d) 1 and 11
23. 23.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
24. 24.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 125 (c) 325 (d) 175
25. 25.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 45 (d) 46
26. 26.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 36 (c) 35 (d) 13
27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
28. 28.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 11 (d) 28
29. 29.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
30. 30.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)
31. 31.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 144 (d) 74
32. 32.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 130 (c) 256 (d) 2
33. 33.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
34. 34.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
35. 35.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 106 (d) 196
36. 36.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 36 (d) 52
37. 37.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
38. 38.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
39. 39.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)
40. 40.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)

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