Question

The number of real or complex solutions of $x^2-6|x|+8=0$ is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The correct option is A

6

$Ifxisreal,x^2-6|x|+8=0\Rightarrow |x{|}^2-6|x|+8=0\Rightarrow |x|=2,4\Rightarrow x=\pm 2,\pm 4Ifxisnon-real,sayx=\alpha +i\beta ,then(\alpha +i\beta {)}^2-6\sqrt{{\alpha }^2+{\beta }^2}+8=0(|\alpha +i\beta |=\sqrt{{\alpha }^2+{\beta }^2})({\alpha }^2-{\beta }^2+8-6\sqrt{{\alpha }^2+{\beta }^2})+2i\alpha \beta =0$

Comparing real and imaginary parts,

$\alpha \beta =0\Rightarrow \alpha =0(if\beta =0thenxisreal.)\&-{\beta }^2+8-6\sqrt{{\beta }^2}=0{\beta }^2\pm 6\beta -8=0\Rightarrow \beta =\frac{\mp 6\pm \sqrt{68}}{2}$

$ie.,\beta =\pm (3-\sqrt{17})Hence\pm (3-\sqrt{17})iarenon-realroots$