Question

Solve the following equation:
$3x^2+15x+2\sqrt{x^2+5x+1}=2$

Answer 1

$3x^2+15x+2\sqrt{x^2+5x+1}=2$

$3(x^2+5x)+2\sqrt{x^2+5x+1}=2$

Let $x^2+5x=t$

then $3t+2\sqrt{t+1}=2$

$4(t+1)=4+9t^2-12t$ (squaring both sides)

$4t+4=4+9t^2-12t$

$9t^2-16t=0$

$t=0$

$x^2+5x=0$

$x=0$ $x=-5$

$t=\frac{16}{9}$

$x^2+5x=\frac{16}{9}$

$9x^2+45x-16=0$

$9x^2+48x-3x-16=0$

$9x^2+48x-3x-16=0$

$3x(3x+16)(3x+16)=0$

$(3x-1)(3x+16)=0$

$x=1/3$ $x=-16/3$