Find the equation and length of common chord of the circle
$\displaystyle
2x^2 + 2y^2 + 7x -5y + 2 = 0
$ and $\displaystyle
x^2 + y^2 - 4x + 8y - 18 =0
$
First, solve for y in each of the equations. Start by moving all x terms and constants to the right side
$\displaystyle 2y^2-5y=-2x^2-7x-2$
complete the square
divide by the coefficient of the quadratic term
$\displaystyle (y-\frac{5}{4})^2=-x^2-\frac{7}{2}x-1+\frac{25}{16}$
taking the root and adding
$\displaystyle y=\pm{\sqrt{-2x^2-\frac{7}{2}x-1+\frac{25}{16}}}+\frac{5}{4}$
Do the same with the other equation and solve the system.
Something wrong here. Those two circles don't intersect at all, so how can they have a common chord? (The centres of the circles are at (2,–4) and $\displaystyle \bigl(-\tfrac74,\tfrac54\bigr)$. The distance between these points is about 6.86. This is a lot more than the sum of the two radii, which are $\displaystyle \sqrt2$ and $\displaystyle \sqrt{58}/4$, both of them less than 2.)
Edit. Oops! First radius is $\displaystyle \sqrt{38}$, not $\displaystyle \sqrt2$. Thanks to skeeter for spotting that.