The number of diagonals in a certain regular polygon is equal to five times the number of sides. How many sides does this polygon have?
Solve for n:
5 n = 1/2 n (n - 3)
Write the quadratic polynomial on the right-hand side in standard form.
Expand out terms of the right-hand side:
5 n = n^2/2 - (3 n)/2
Move everything to the left-hand side.
Subtract n^2/2 - (3 n)/2 from both sides:
(13 n)/2 - n^2/2 = 0
Factor the left-hand side.
Factor n and constant terms from the left-hand side:
-1/2 n (n - 13) = 0
Multiply both sides by a constant to simplify the equation.
Multiply both sides by -2:
n (n - 13) = 0
Find the roots of each term in the product separately.
Split into two equations:
n - 13 = 0 or n = 0
Look at the first equation: Solve for n.
Add 13 to both sides:
n = 13 or n = 0(Discard) . So this is a 13-gon polygon with 65 diagonals.