Question

✓20%7D%20%20%3D%205" id="TexFormula1" title=" \sqrt{19 + \sqrt{30 + \sqrt{32 + x } } } = 5" alt=" \sqrt{19 + \sqrt{30 + \sqrt{32 + x } } } = 5" align="absmiddle" class="latex-formula">
can someone please solve this​

Answer 1

We'll have to repeatedly square both sides of the equation, in order to get rid of the square roots. Squaring a first time yields

$19+\sqrt{30+\sqrt{32+x}}=25$

Move the 19 to the right hand side:

$\sqrt{30+\sqrt{32+x}}=6$

And square again:

$30+\sqrt{32+x}=36 \iff \sqrt{32+x}=6$

Square one last time:

$32+x=36 \iff x=36-32=4$

Let's check the solutions: all these squaring might have created external solutions:

$\sqrt{19+\sqrt{30+\sqrt{32+4}}}=\sqrt{19+\sqrt{30+6}}=\sqrt{19+6}=\sqrt{25}=5$

So, $x=4$ is a feasible solution.