Question

If a ≤ 0, the roots of $x^2-2a|x-a|-3a^2=0$ is / are :

• A. a
• B. a
• C.
• D.

Answer 1

The correct option is D

Let a = -p, then p $\ge$ 0. Then equation becomes

$x^2+2p|x+p|-3p^2=0$ . ( p $\ge$ 0 ).

Case I : $x<-p$ . Then

$x^2-2px-5p^2=0$ $\Rightarrow$ $x=\frac{(2p\pm \sqrt{24p^2})}{2}$

$\Rightarrow$ x = p $\pm$ $\sqrt{6}p$ $\Rightarrow$ x = p (1 + $\sqrt{6}$) or i.e., ( $\sqrt{6}-1)a.$

Case II : x $\ge$ -p. Then $x^2+2px-p^2=0$

$\Rightarrow$ x = $\frac{(-2p\pm \sqrt{8p^2})}{2}=-p\pm \sqrt{2}$ p

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

$Asx\ge -p,x=(-1+\sqrt{2})p=(-1-\sqrt{2})p$

Hence solutions are $(\sqrt{6}-1)aand(1-\sqrt{2})a$