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1. $(4 + \sqrt{7}),$ expressed as a perfect square, is equal to = ?

A. ${(2 + \sqrt{7})}^{2}$

B. ${(\frac{\sqrt{7}}{2} + \frac{1}{2})}^{2}$

C. ${\frac{1}{2} {(\sqrt{7} + 1)}^{2}}$

D. $(\sqrt{3} + \sqrt{4})$

Solution:

\begin{aligned} (4 + \sqrt{7}) \\ = \frac{7}{2} + \frac{1}{2} + 2 \times \frac{\sqrt{7}}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \\ = {(\frac{\sqrt{7}}{\sqrt{2}})}^{2} + {(\frac{1}{\sqrt{2}})}^{2} + 2 \times \frac{\sqrt{7}}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \\ = {(\frac{\sqrt{7}}{\sqrt{2}} + \frac{1}{\sqrt{2}})}^{2} \\ = \frac{1}{2} {(\sqrt{7} + 1)}^{2} \end{aligned}

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