Question

Find the mean proportional between
$\left(i\right)$ $6+3\sqrt{3}\text{and}8-4\sqrt{3}$

Answer 1

Step: Find the mean proportional

$6+3\sqrt{3}\text{and}8-4\sqrt{3}$

Let mean proportional be $x$

$\therefore 6+3\sqrt{3},x$ and $8-4\sqrt{3}$ are in proportional

$\Rightarrow \frac{6+3\sqrt{3}}{x}=\frac{x}{8-4\sqrt{3}}$

$\Rightarrow x\times x=(6+3\sqrt{3})(8-4\sqrt{3})$

$\Rightarrow x^2=48-24\sqrt{3}+24\sqrt{3}-12(\sqrt{3}{)}^2$

​​​​​​​$\Rightarrow x^2=48-12\times 3$

​​​​​​​$\Rightarrow x^2=48-36$

$\Rightarrow x^2=12$

$\Rightarrow x=\sqrt{12}$

$\therefore x=2\sqrt{3}$

Hence, the mean proportional is $2\sqrt{3}$