absolute value in a quadratic

Question

If $a<-2$ is a real number, then the equation:

$x^2+a|x|+1=0$

has how many real roots?

After finding the roots in terms of $a$, how do I proceed?

Answer 1

Consider two cases, whether the roots satisfies $x\ge0$ or $x<0$.

For the first case, $|x| = x$, so the equation becomes $x^2+ax+1 = 0$, $$\text{Discriminant} = a^2 - 4 > (-2)^2-4 = 0$$ This shows that the equation $x^2+ax+1 = 0$ alone has two real roots, but we still have to check whether the two roots are in the $x\ge0$ range: $$\text{Sum of roots} = -a > 2>0\\\text{Product of roots} = 1>0$$ These shows the two roots sum to a positive number, and they are of the same sign; i.e. they are both positive, and fits our first case.

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The second case can be done similarly, or by noticing $x^2+a|x|+1$ is an even function...