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

Concept & Formula To convert an expression into a perfect square, we often need to introduce a $2$ for the $2ab$ term in the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$. If no $2$ is present before the radical, multiply and divide the entire expression by $2$. Step-by-Step Solution * Given expression: $4 + \sqrt{7}$ * Multiply and divide the expression by $2$ to create the $2ab$ term: $$ \frac{1}{2} (8 + 2\sqrt{7}) $$ * Now, focus on the numerator $8 + 2\sqrt{7}$. We need two numbers that multiply to $7$ and add to $8$. These are $7$ and $1$. * Rewrite the numerator: $$ 8 + 2\sqrt{7} = 7 + 1 + 2\sqrt{7} = (\sqrt{7})^2 + (1)^2 + 2(\sqrt{7})(1) $$ * This condenses into a perfect square: $$ (\sqrt{7} + 1)^2 $$ * Substitute this back into the fraction: $$ \frac{1}{2} (\sqrt{7} + 1)^2 $$ Exam Strategy & Shortcut Instead of expanding every option, you can quickly evaluate them by approximation. $\sqrt{7}$ is approx $2.64$. So $4 + 2.64 = 6.64$. Option (c) is $0.5 \times (2.64 + 1)^2 = 0.5 \times (3.64)^2 \approx 0.5 \times 13.25 = 6.62$. Option (c) is the match. Alternatively, recognizing the missing $2$ for the $2ab$ term immediately points towards multiplying/dividing by $2$, which is exclusively present in Option (c). Common Pitfall A very common mistake is squaring the expression itself instead of expressing it *as* a square. Students might mistakenly square $(4 + \sqrt{7})$ yielding $16 + 7 + 8\sqrt{7}$, which doesn't help solve the question. Final Answer Therefore, the correct answer is $\left\{ \frac{1}{2}(\sqrt{7} + 1)^2 \right\}$.