# The number of solutions to the equation $|x|^2+3|x|+2=0$ is

$\begin{array}{1 1}(A) \; 0 \\(B)\; 1 \\(C)\; 2 \\(D)\; 4 \end{array}$

Toolbox:
• If $ax^2+bx+c=0$, then $x=\large\frac{-b\pm\sqrt {b^2-4ac}}{2a}$
• If $D=b^2-4ac\geq 0$ then the values of x are real
• If $D=b^2-4ac< 0$ then the values of x are complex
• $|z|$ is always a positive real number regardless of $x$ being a real number or complex number.
Given eqn. is $|x|^2+3|x|+2=0$
$\Rightarrow\:\:|x|=\large\frac{-3\pm\sqrt {9-8}}{2}$$=-2\:\:or\:-1$
But $|x|$ cannot be negative
$\therefore$ No solution exists for this equation.