Answer
The solutions are $x=-7$, $x=1$ and $x=-3$
Work Step by Step
$|x^{2}+6x+1|=8$
Solving an absolute value equation is equivalent to solving two separate equations. In this case, the equations are $x^{2}+6x+1=8$ and $x^{2}+6x+1=-8$
Solve the first equation:
$x^{2}+6x+1=8$
Take $8$ to the left side and simplify:
$x^{2}+6x+1-8=0$
$x^{2}+6x-7=0$
Solve by factoring:
$(x+7)(x-1)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x+7=0$
$x=-7$
$x-1=0$
$x=1$
Solve the second equation:
$x^{2}+6x+1=-8$
Take $8$ to the left side and simplify:
$x^{2}+6x+1+8=0$
$x^{2}+6x+9=0$
Solve by factoring:
$(x+3)^{2}=0$
Take the square root of both sides:
$\sqrt{(x+3)^{2}}=\sqrt{0}$
$x+3=0$
Solve for $x$:
$x=-3$
The solutions are $x=-7$, $x=1$ and $x=-3$
