Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 939 · 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 are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 4 and 9 (d) 2 and 18
2. 2.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 277 (d) 529
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 244 (d) 484
5. 5.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 52 (d) 36
6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
7. 7.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)
8. 8.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
9. 9.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
10. 10.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
11. 11.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 35 (d) 13
12. 12.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 37 (c) 43 (d) 61
13. 13.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2916 (d) 2516
14. 14.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 8 (d) 18
15. 15.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 7931 (d) 8099
16. 16.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
18. 18.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
20. 20.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 3 and 8 (d) 1 and 10
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
22. 22.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
23. 23.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 2 (c) 74 (d) 144
24. 24.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 121 (d) 169
25. 25.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
26. 26.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 9 (c) 21 (d) 20
27. 27.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 14 (d) 47
28. 28.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
29. 29.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 10816 (d) 9216
30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
31. 31.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 2 (d) 16
32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
33. 33.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 137 (c) 225 (d) 181
34. 34.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
35. 35.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 30 (d) 31
36. 36.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
38. 38.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
39. 39.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 1 (d) 61
40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 244 (d) 172

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