Question

If $a,b,c$ are three distinct positive real numbers then the number of real roots $a{x}^2+2b|x|-c=0$ is

• A. $4$
• B. $2$
• C. $0$
• D. None of the above.

Answer 1

The correct option is B $2$

a,b,c are $3$ distinct positive real numbers

$a{x}^2+2b\left|x\right|-c=0$

if $x<0$

$a{x}^2-2bx-x=0$

$x=\frac{2b\pm \sqrt{4b^2+4ac}}{2a}$

$\therefore x=\frac{b-\sqrt{b^2+ac}}{a}(\because x<0)$

$\therefore x\ne \frac{b+\sqrt{b^2+ac}}{a}>0$

if $x>0$

$a{x}^2+2bx-c=0$

$x=\frac{-2b\pm \sqrt{4b^2+4ac}}{2a}$

$x=\frac{-b\pm \sqrt{b^2+ac}}{a}$

$x=\frac{-b+\sqrt{b^2+ac}}{a}$

As $x>0,x\ne \frac{-b-\sqrt{b^2+ac}}{a}$

$\therefore$There are $2$ real roots for $a{x}^2+2b\left|x\right|-c=0$