+26367 If \(\sqrt{5 + x} + \sqrt{20 - x} = 7\), what is the value of \((5 + x)(20 - x)\)?
\(\begin{array}{|rcll|} \hline \sqrt{5 + x} + \sqrt{20 - x} &=& 7 \quad &|\quad \text{square both sides} \\ \left(\sqrt{5 + x} + \sqrt{20 - x}\right)^2 &=& 7^2 \\ \left(\sqrt{5 + x}\right)^2 +2\sqrt{5 + x} \sqrt{20 - x} + \left(\sqrt{20 - x}\right)^2 &=& 49 \\ 25 +2\sqrt{(5 + x)(20 - x)} &=& 49 \quad &|\quad -25 \\ 2\sqrt{(5 + x)(20 - x)} &=& 49-25 \\ 2\sqrt{(5 + x)(20 - x)} &=& 24 \quad &|\quad : 2 \\ \sqrt{(5 + x)(20 - x)} &=& 12 \quad &|\quad \text{square both sides} \\ \left( \sqrt{(5 + x)(20 - x)}\right)^2 &=& 12^2 \\ \mathbf{(5 + x)(20 - x)} &=& \mathbf{144} \\ \hline \end{array}\)
