Quadratic Equation with $ |x|$

Question

Hello so I am trying to solve this problem:

$2x^2 - (a-4)|x| - a + 10 = 0 \ $ calculate every $a$, in which this equation has only $2$ solutions.

I tried to check two examples

1. $|x| \ge0$
2. $|x| < 0$
   

but it should have $2$ solutions in total. I don't know what to do. Can anyone help me out?

Answer 1

Sometimes you just have to know which rocks to look under.

Since the eqation is unchanged by replacing $x$ with $-x$, we are sure of exactly two roots when there is exactly one positive root (because the other root is the additive inverse) and no zero root.

A quadratic equation will have one positive root on two cases:

(1) when the $x^2$ coefficient ($2$, in this case) and the constant term ($-a+10$) have opposite signs.

(2) when the discriminant is zero, so you have a double root when the absolute value sign is removed, and this double root is positive. Note that a positive double root requires the linear term to have a coefficient with sign opposite the quadratic term coefficient, aong with the zero discriminant.

Finish from there.

Answer 2

Hint:

$|x|=\begin{cases}x, \ x \ge 0,\\ -x, \ x < 0.\end{cases}$