Question

The values of $x$ which satisfy the equation $\sqrt{{5x}^2-8x+3}-\sqrt{{5x}^2-9x+4}=\sqrt{({2x}^2-2x)}-\sqrt{({2x}^2-3x+1)}$ is

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is C $1$

Instead of squaring and complicating, we can check the result by substituting the values of x, preferably 0,1 and -1.
By substituting $x=1$ we get $LHS=RHS=0$
Hence $x=1$ is a solution for the above equation.
Now
$5x^2-8x+3\ge 0$
$5x^2-5x-3x+3\ge 0$
$(5x-3)(x-1)>0$
Hence
$x\ge 1$ and $x\le \frac{3}{5}$
Similarly
$5x^2-9x+4\ge 0$
$(5x-4)(x-1)\ge 0$
$x\ge 1$ and $x\le \frac{4}{5}$
$2x^2-2x\ge 0$
$2x(x-1)\ge 0$
$x\ge 1$ and $x\le 0$
and
$2x^2-3x+1\ge 0$
$2x^2-2x-x+1\ge 0$
$(2x-1)(x-1)\ge 0$
$x\ge 1$ and $x\le \frac{1}{2}$
Hence there is only one solution that is $x=1$