Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 978 · 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 product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
3. 3.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
4. 4.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 109 (c) 169 (d) 7
5. 5.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
6. 6.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
7. 7.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
8. 8.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 289 (d) 217
9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
10. 10.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 81 (c) 121 (d) 101
11. 11.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
12. 12.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3969 (d) 3591
13. 13.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
14. 14.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 19 (c) 9 (d) 17
15. 15.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 10 (d) 21
16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
18. 18.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 7 (c) 10 (d) 9
19. 19.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
20. 20.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 16 (d) 2
21. 21.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
22. 22.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 193 (d) 277
23. 23.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
24. 24.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
25. 25.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
26. 26.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 7 (c) 8 (d) 9
27. 27.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)
28. 28.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
29. 29.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
33. 33.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 14 (d) 12
34. 34.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2809 (d) 2209
35. 35.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 5 (c) 81 (d) 53
36. 36.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 121 (d) 65
37. 37.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
38. 38.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 208 (c) 144 (d) 112
39. 39.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
40. 40.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256

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