Question

Find the mean proportional between `6 + 3sqrt(3)` and `8 - 4sqrt(3)`

Answer 1

Let the mean proportional between `6 + 3sqrt(3)` and `8 - 4sqrt(3)` be x.

`\implies 6 + 3sqrt(3), x` and `8 - 4sqrt(3)` are in continued proportion.

`\implies 6 + 3sqrt(3) : x = x : 8 - 4sqrt(3)`

`\implies x xx x = (6 + 3sqrt(3)) (8 - 4sqrt(3))`

`\implies x^2 = 48 + 24sqrt(3) - 24sqrt(3) - 36`

`\implies x^2 = 12`

`\implies x = 2sqrt(3)`