A point in a quadrilateral

The first step is to solve for $x$ , where $-2x+10.5 = x^2 - 4x + 7.5$ . The solutions are $-1$ and $3$ , but $-1$ is negative, so $3$ is the solution. After substituting the value $3$ into $(2x+1)$ for the length of $CE$ , a system of equations can be formed using the Pythagorean Theorem relating $BD$ , $DC$ , and $AD$ , Stewart's Theorem relating $BD$ , $AB$ , $AE$ , $ED$ , and $BE$ , and the British Flag theorem relating $CE$ , $BE$ , $AE$ , and $ED$ . These equations can be used to determine $BE$ , $ED$ , and $BD$ , where the values can be added to determine the perimeter of triangle $BED$ for a final answer of $18.764$ .