Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 385 · 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 = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 112 (d) 208
2. 2.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 58 (c) 79 (d) 142
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
4. 4.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 184 (d) 144
5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
6. 6.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 53 (d) 15
7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2601 (d) 2501
9. 9.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
11. 11.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
14. 14.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 4 and 7 (d) 1 and 10
15. 15.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
16. 16.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 185 (d) 537
17. 17.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)
18. 18.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 27 (d) 28
19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
20. 20.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 266 (c) 384 (d) 576
21. 21.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 34 (c) 64 (d) 2
22. 22.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 1 and 13 (c) 3 and 15 (d) 5 and 9
23. 23.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
24. 24.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
25. 25.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4901 (d) 4761
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
28. 28.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 3 and 4 (c) 1 and 10 (d) 1 and 6
29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 361 (d) 193
30. 30.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3584 (d) 3146
31. 31.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6
32. 32.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
34. 34.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
35. 35.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
36. 36.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7396 (c) 7406 (d) 8084
37. 37.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
39. 39.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 157 (d) 5
40. 40.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 2 and 27 (d) 1 and 54

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