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  4. The number of roots of the equation √x2-4-(x-2)=√x2-5 x+6 is

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Q. The number of roots of the equation $\sqrt{x^2-4}-(x-2)=\sqrt{x^2-5 x+6}$ is

Complex Numbers and Quadratic Equations

Solution:

$\sqrt{x^2-4}-(x-2)=\sqrt{x^2-5 x+6}$
$\sqrt{x-2}(\sqrt{x+2}-\sqrt{x-2})=\sqrt{x-2} \sqrt{x-3}$
So $x=2$ or $\sqrt{x+2}-\sqrt{x-2}=\sqrt{x-3}$
squaring both side
$x+2+x-2-2 \sqrt{x^2-4}=x-3$
$x+3=2 \sqrt{x^2-4}$
$x^2+6 x+9=4 x^2-16$
$\Rightarrow 3 x^2=6 x-25=0$
$D>0$