$\displaystyle x^2+y^2=4$

$\displaystyle 16x^2+9y^2=144$

The answer is not as important as how you arrived there, wolfram alpha can compute these. I would like to know how to solve manually.

I keep getting the equation $\displaystyle 25x^2-36x-108$ from substitution(solved for formula 1, subbed into 2),

elimination yields nothing because there are no real roots, the graph is a circle within an ellipse.

Using the quadratic formula with the above equation gives me much different answers than wolfram.

I get $\displaystyle \frac{36\pm \sqrt{12096}}{50}$ Which does not make much sense.