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1. If $\frac{(x - \sqrt{24}) (\sqrt{75} + \sqrt{50})}{\sqrt{75} - \sqrt{50}}$ = 1 then the value of x is = ?

A. $\sqrt{5}$

B. 5

C. $2 \sqrt{5}$

D. $3 \sqrt{5}$

Solution:

\begin{aligned} \frac{(x - \sqrt{24}) (\sqrt{75} + \sqrt{50})}{\sqrt{75} - \sqrt{50}} = 1 \\ \Rightarrow (x - \sqrt{24}) = \frac{\sqrt{75} - \sqrt{50}}{\sqrt{75} + \sqrt{50}} \\ \Rightarrow (x - \sqrt{24}) = \frac{{(\sqrt{75} - \sqrt{50})}^{2}}{75 - 50} \\ \Rightarrow (x - \sqrt{24}) = \frac{75 + 50 - 2 \sqrt{75} \sqrt{50}}{25} \\ \Rightarrow (x - \sqrt{24}) = \frac{125 - 2 \times 5 \sqrt{3} \times 5 \sqrt{2}}{25} \\ \Rightarrow (x - \sqrt{24}) = \frac{125 - 50 \sqrt{6}}{25} \\ \Rightarrow (x - \sqrt{24}) = \frac{25 (5 - 2 \sqrt{6})}{25} \\ \Rightarrow x - 2 \sqrt{6} = 5 - 2 \sqrt{6} \\ \Rightarrow x = 5 \end{aligned}

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