Question

The number of real solutions $\sqrt{({3x}^2-7x-30)}-\sqrt{({2x}^2-7x-5)}=x-5$ is

• A. one
• B. two
• C. three
• D. none of these

Answer 1

The correct option is A one

$\sqrt{3x^2-7x-30}-\sqrt{2x^2-7x-5}=x-5$
$\Rightarrow \sqrt{3x^2-7x-30}=\sqrt{2x^2-7x-5}+(x-5)$
$squaringbothsides$
$3x^2-7x-30={(x-5)}^2+2x^2-7x-5+2(x-5)\sqrt{2x^2-7x-5}$
$\Rightarrow -7x+17x-30-20-2(x-5)\sqrt{2x^2-7x-5}=0$
$\Rightarrow 10(x-5)=2(x-5)\sqrt{2x^2-7x-5}$
$\Rightarrow 5=\sqrt{2x^2-7x-5}$
$squaringbothsides$
$25=2x^2-7x-5$
$\Rightarrow 2x^2-7x-30=0$
$\Rightarrow (2x+5)(x-6)=0$
$x=\frac{-5}{2},6$
$forx=\frac{-5}{2}$
$norealroots$
$forx=6$
$onlyonesolution$