What does it mean to solve the equation of an ellipse as a quadratic? An illustration required me to find out the area of the curve $$5x^2 + 6xy + 2y^2 + 7x + +6y + 6 = 0 $$
 
They proceeded to solve the equation as a quadratic obtaining $y_1$ and $y_2$ as the two branches.   
Why are $y_1$ and $y_2$ as depicted in the diagram?  
 if it isn't too broad what does it mean to solve a quadratic equation for any conic section (parabola, ellipse, hyperbola, circle. 

The point of this is to write $y$ as a function of $x$ - well, actually, two functions. So, then, we treat $x$ as a constant and sort everything by powers of $y$:
\begin{align*} 0 &= 5x^2+6xy+2y^2+7x+6y+6\\
0 &= 2y^2 + (6xy+6y) + (5x^2+7x+6)\\
0 &= 2y^2 + (6x+6)y + (5x^2+7x+6)\end{align*}
Since we're treating $x$ as a constant, that's just a quadratic equation in $y$ - and we know how to solve quadratic equations:
\begin{align*}y &= \frac{-(6x+6)\pm\sqrt{(6x+6)^2-4\cdot 2\cdot (5x^2+7x+6)}}{2\cdot 2}\\
&= \frac{-6(x+1)\pm\sqrt{36x^2+72x+36-40x^2-56x-48}}{4}\\
&= \frac{-6(x+1)\pm\sqrt{-4x^2+16x-12}}{4}\\
y &= \frac{-3(x+1)\pm\sqrt{(x-1)(3-x)}}{2}\end{align*}
There it is. That quadratic equation has a $\pm$ sign in it, and both choices for the square root are viable. If we take the $+$ sign, we get a larger value of $y$, for the top curve $y_1$ drawn in pink. If we take the $-$ sign, we get a smaller value of $y$, for the bottom curve $y_2$ drawn in green.
Similar reasoning applies to any conic section; if we treat one variable as a constant, the equation becomes a linear or quadratic equation in the remaining variable, which we can solve explicitly.
