Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 821 · 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 = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
2. 2.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
3. 3.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 85 (d) 127
4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
5. 5.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
6. 6.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 36 (c) 13 (d) 35
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
8. 8.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
9. 9.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175
10. 10.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
16. 16.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
17. 17.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
18. 18.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
19. 19.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
20. 20.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
21. 21.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
22. 22.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3731 (d) 3481
23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
24. 24.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
25. 25.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
26. 26.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 37 (d) 49
27. 27.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 9216 (d) 10816
28. 28.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
30. 30.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
31. 31.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 5 and 6 (d) 1 and 30
32. 32.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 19 (d) 11
33. 33.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 10211 (d) 9801
34. 34.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 153 (d) 9
35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
36. 36.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 324 (d) 244
37. 37.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 16 (d) 4
38. 38.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 313 (d) 137
39. 39.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
40. 40.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 82 (c) 8 (d) 100

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