Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 192 · 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 = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 40 (d) 4
2. 2.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 5329 (d) 4489
3. 3.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
4. 4.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
5. 5.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 136 (d) 256
6. 6.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
7. 7.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
8. 8.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
9. 9.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)
10. 10.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
11. 11.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
14. 14.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3249 (d) 3259
15. 15.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
16. 16.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
17. 17.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
18. 18.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 6 (d) 8
19. 19.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
20. 20.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 196 (d) 100
21. 21.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
24. 24.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 276 (d) 196
25. 25.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
26. 26.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3364 (c) 3596 (d) 3844
27. 27.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
28. 28.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
29. 29.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 262 (d) 163
30. 30.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)
31. 31.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
33. 33.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
34. 34.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 33 (d) 32
35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 15 (d) 63
36. 36.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
37. 37.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)
38. 38.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 4901 (d) 5051
39. 39.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 6384 (c) 5776 (d) 5786
40. 40.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)

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