Rewrite \(36-{x}^{2}\) in the form \({a}^{2}-{b}^{2}\), where \(a=6\) and \(b=x\).
\[\sqrt{122}-\sqrt{{6}^{2}-{x}^{2}}=13\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\sqrt{122}-\sqrt{(6+x)(6-x)}=13\]
Separate terms with roots from terms without roots.
\[-\sqrt{(6+x)(6-x)}=13-\sqrt{122}\]
Square both sides.
\[(6+x)(6-x)=291-26\sqrt{122}\]
Expand.
\[{6}^{2}-{x}^{2}=291-26\sqrt{122}\]
Simplify \({6}^{2}\) to \(36\).
\[36-{x}^{2}=291-26\sqrt{122}\]
Subtract \(36\) from both sides.
\[-{x}^{2}=291-26\sqrt{122}-36\]
Simplify \(291-26\sqrt{122}-36\) to \(255-26\sqrt{122}\).
\[-{x}^{2}=255-26\sqrt{122}\]
Multiply both sides by \(-1\).
\[{x}^{2}=-255+26\sqrt{122}\]
Take the square root of both sides.
\[x=\pm \sqrt{-255+26\sqrt{122}}\]
Check solution
When \(x=\sqrt{-255+26\sqrt{122}}\), the original equation \(\sqrt{122}-\sqrt{{6}^{2}-{x}^{2}}=13\) does not hold true.We will drop \(x=\sqrt{-255+26\sqrt{122}}\) from the solution set.
Check solution
When \(x=-\sqrt{-255+26\sqrt{122}}\), the original equation \(\sqrt{122}-\sqrt{{6}^{2}-{x}^{2}}=13\) does not hold true.We will drop \(x=-\sqrt{-255+26\sqrt{122}}\) from the solution set.
