Quadratic equation with absolute value, is there a real number for which equation has unique solution

Question

Problem :

Whether there is a real number a, for which the equation $x^2+|x|+a = 0$ has a unique solution in the set of real numbers?

I tried to discuss the solution of this equation, but I can not get to that spot.

Answer 1

If $x \ge 0$ then $x = \frac {-1\pm \sqrt{1- 4a}}2$ but only $\frac {-1 + \sqrt{1-4a}}2 \ge 0$.

If $x < 0$ then $x = \frac {1- \sqrt{1-4a}}2$

So only way there is a single unique solution is if $1 - \sqrt{1-4a} = -1 + \sqrt{1-4a}$ and $\sqrt {1-4a} = 1$ so $a = 0$. will have a unique solution $x$ and no other $a$ will.

... or ....

$(-x)^2 + |-x| + a = x^2 +|x| + a$ so if $x$ is a solution then so is $-x$. So to be unique $x = 0$ must be the only solution.

So $0^2 + |0| + a= 0$.

So $a = 0$ is only option.