How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Question

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ^2L^2-B^2=0$.

Using the formula for quadratic equations:

$x = \frac{2(\epsilon^2L-B)\pm \sqrt{(B-\epsilon ^2L)^2-4(\epsilon ^2-1)(\epsilon ^2L^2-B^2)}}{2(\epsilon ^2-1)}$

This is supposed to give:

$x = \frac{B\pm\epsilon L}{1\pm \epsilon}$,

but, how? It's easy to verify the solution by insertion, but how do I solve the equation?

Answer 1

You have done an error in your discriminant which should begin by a "4":

$$\Delta=4(B-\epsilon ^2L)^2-4(\epsilon ^2-1)(\epsilon ^2L^2-B^2)$$

Expanding it, and then factoring it, you will obtain the following result:

$$\Delta = 4 \epsilon^2 (B-L)^2$$

which is a perfect square ; then you will find the (simple) desired formula for $x$.

Answer 2

If you use for $ax^2+bx+c=0$ $$\Delta=b^2-4ac$$ expand and simplify; you should get $$\Delta=4 \epsilon ^2 (B-L)^2$$ I am sure that you can take it from here.