Question
Solve the following equation by reducing it to quadratic equation:
$\sqrt{3 x^2-2}+1=2 x$.
$\sqrt{3 x^2-2}+1=2 x$.
✓
Answer
$\sqrt{3 x^2-2}+1=2 x$
$\Rightarrow \sqrt{3 x^2-2} $
$ =2 x-1$
On squaring both sides, we get
$3 x ^2-2=4 x ^2+1-4 x $
$\Rightarrow- x ^2+4 x -3=0 $
$\Rightarrow x ^2-4 x +3=0$
$\Rightarrow x ^2-3 x - x +3=0 $
$\Rightarrow x ( x -3)-1( x -3)=0$
$ \Rightarrow( x -3)( x -1)=0$
$ \Rightarrow x =3 \text { or } x =1$
Hence, the solutions are $[3, 1].$
$\Rightarrow \sqrt{3 x^2-2} $
$ =2 x-1$
On squaring both sides, we get
$3 x ^2-2=4 x ^2+1-4 x $
$\Rightarrow- x ^2+4 x -3=0 $
$\Rightarrow x ^2-4 x +3=0$
$\Rightarrow x ^2-3 x - x +3=0 $
$\Rightarrow x ( x -3)-1( x -3)=0$
$ \Rightarrow( x -3)( x -1)=0$
$ \Rightarrow x =3 \text { or } x =1$
Hence, the solutions are $[3, 1].$
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