Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 17 · 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.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)
2. 2.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
3. 3.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
4. 4.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 1225 (d) 635
5. 5.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
6. 6.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 1 (d) 9
7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 1625 (d) 2035
8. 8.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 4904 (d) 5194
9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 273 (d) 537
10. 10.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
11. 11.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
12. 12.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)
13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
14. 14.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 2 and 5 (d) 1 and 8
15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
16. 16.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 484 (d) 324
17. 17.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
18. 18.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
19. 19.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 10619 (d) 9409
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
22. 22.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 97 (c) 169 (d) 241
23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
24. 24.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
26. 26.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
27. 27.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
28. 28.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 64 (c) 8 (d) 144
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
30. 30.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
31. 31.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
32. 32.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 94 (d) 49
33. 33.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 625 (d) 225
34. 34.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
35. 35.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 10 (d) 11
36. 36.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 52
37. 37.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
38. 38.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 109 (c) 7 (d) 49
39. 39.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
40. 40.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 100 (d) 148

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