Question

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{2d}{3}}
b. n = \sqrt{2d+3}
c. n = \sqrt{2d}+3
d. n = \sqrt{2d+\frac{9}{4}}+\frac{3}{2}

Explanation:

Step1: Start with given formula

$d=\frac{n(n - 3)}{2}$

Step2: Cross - multiply

$2d=n^2-3n$

Step3: Complete the square

$2d+\frac{9}{4}=n^2-3n+\frac{9}{4}=(n - \frac{3}{2})^2$

Step4: Solve for n

$n=\sqrt{2d+\frac{9}{4}}+\frac{3}{2}$

Answer:

D. $n=\sqrt{2d+\frac{9}{4}}+\frac{3}{2}$