Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 136 · 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.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
2. 2.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 144 (d) 64
3. 3.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 3 (c) 2 and 4 (d) 1 and 6
4. 4.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 2 and 9 (d) 1 and 10
5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
6. 6.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
7. 7.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 35 (d) 34
8. 8.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 208 (d) 592
10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
11. 11.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 184 (d) 144
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
13. 13.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
16. 16.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 12 (d) 11
17. 17.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2601 (d) 2401
18. 18.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
19. 19.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
20. 20.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
21. 21.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 225 (d) 375
22. 22.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
23. 23.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 10 (d) 11
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
26. 26.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
27. 27.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2819 (d) 2509
28. 28.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
31. 31.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
32. 32.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 256 (c) 136 (d) 4
33. 33.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)
34. 34.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 12 (d) 8
35. 35.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 3 (d) 9
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
37. 37.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)
38. 38.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 2 and 14 (d) 1 and 28
39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
40. 40.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 289 (d) 3

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