Question

The sum of the roots of the equation $\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$ is

• A. $\frac{5}{6}$
• B. $\frac{6}{5}$
• C. $\frac{7}{3}$
• D. $\frac{3}{7}$

Answer 1

The correct option is B $\frac{6}{5}$

Let $5x^2-6x=y$
Then,
$\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$
$\Rightarrow \sqrt{y+8}-\sqrt{y-7}=1$
For the square root to exist,
$y+8\ge 0\Rightarrow y\ge -8y-7\ge 0\Rightarrow y\ge 7\therefore y\ge 7$

$\Rightarrow \sqrt{y+8}=1+\sqrt{y-7}$
Squaring on both the sides
$\Rightarrow y+8=1+y-7+2\sqrt{y-7}\Rightarrow \sqrt{y-7}=7$
Squaring on both the sides,
$\Rightarrow y-7=49\Rightarrow y=56$
Now,
$\Rightarrow 5x^2-6x=56\Rightarrow 5x^2-6x-56=0$

Therefore, the sum of roots $=\frac{6}{5}$