Question

The sum of the roots of the equation, $x^2-|2x-3|-4=0$ is

• A. $2$
• B. $-2$
• C. $\sqrt{2}$
• D. $-\sqrt{2}$

Answer 1

The correct option is D $-\sqrt{2}$

$x^2-|2x-3|-4=0$

$|2x-3|=2x-3$, when $2x-3>0\Rightarrow x>\frac{3}{2}$

For the above case, the equation becomes $x^2-2x-1=0$

$\Rightarrow x=\frac{(2\pm \sqrt{8})}{2}=(1\pm \sqrt{2})$

$\Rightarrow x=(1+\sqrt{2})$

$1-\sqrt{2}$ is discarded since $x$ should be greater than $\frac{3}{2}$

Now, for $x<\frac{3}{2},|2x-3|=3-2x$

The equation becomes $x^2+2x-7=0$

$x=\frac{-2\pm \sqrt{4+7\times 4}}{2}=\frac{-2\pm 2\sqrt{8}}{2}=-1\pm 2\sqrt{2}\Rightarrow$ only $-1-2\sqrt{2}$

$-1+2\sqrt{2}$ is discarded because it's more than $\frac{3}{2}$

Sum of the roots obtained is=$-1-2\sqrt{2}+1+\sqrt{2}=-\sqrt{2}$.