Question

Solve the equation:
$\frac{\sqrt{4x+1}+\sqrt{x+3}}{\sqrt{4x+1}-\sqrt{x+3}}=\frac{4}{1}$

Answer 1

Consider the given equation.

$\frac{\sqrt{4x+1}+\sqrt{x+3}}{\sqrt{4x+1}-\sqrt{x+3}}=\frac{4}{1}$

$\frac{\sqrt{4x+1}+\sqrt{x+3}}{\sqrt{4x+1}-\sqrt{x+3}}\times \frac{\sqrt{4x+1}+\sqrt{x+3}}{\sqrt{4x+1}+\sqrt{x+3}}=\frac{4}{1}$

$\frac{{(\sqrt{4x+1}+\sqrt{x+3})}^2}{{\left(\sqrt{4x+1}\right)}^2-{\left(\sqrt{x+3}\right)}^2}=\frac{4}{1}$

$\frac{4x+1+x+3+2\sqrt{(4x+1)(x+3)}}{(4x+1)-(x+3)}=\frac{4}{1}$

$\frac{5x+4+2\sqrt{4x^2+12x+x+3}}{4x+1-x-3}=\frac{4}{1}$

$\frac{5x+4+2\sqrt{4x^2+13x+3}}{3x-2}=\frac{4}{1}$

$5x+4+2\sqrt{4x^2+13x+3}=12x-8$

$2\sqrt{4x^2+13x+3}=7x-12$

$4(4x^2+13x+3)=49x^2+144-168x$

$16x^2+52x+12=49x^2-168x+144$

$33x^2-220x+132=0$

$x=\frac{220\pm \sqrt{{\left(220\right)}^2-4\times 33\times 132}}{66}$

$x=\frac{220\pm \sqrt{48400-17424}}{66}$

$x=\frac{220\pm \sqrt{30976}}{66}$

$x=\frac{220\pm 176}{66}$

$x=\frac{220+176}{66},\frac{220-176}{66}$

$x=6,\frac{2}{3}$

Hence, the value of $x$ is $6,\frac{2}{3}$.