Problem
The number of diagonals in a polygon, $d$, in terms of the number of sides of a polygon, $n$, is given by the equation $d=\frac{n(n-3)}{2}$. What equation shows the number of sides of a polygon in terms of the number of diagonals A. $n=\sqrt{\frac{2 d}{3}}$ B. $n=\sqrt{2 d+3}$ c. $n=\sqrt{2 d}+3$ D. $n=\sqrt{2 d+\frac{9}{4}}+\frac{3}{2}$
Solution
Step 1 ：Given the equation for the number of diagonals in terms of the number of sides: \(d=\frac{n(n-3)}{2}\)
Step 2 ：Solve for n using the quadratic formula: \(n = \frac{3 \pm \sqrt{9 + 8d}}{2}\)
Step 3 ：The final answer is: \(\boxed{n=\sqrt{2d+\frac{9}{4}}+\frac{3}{2}}\)
