| x^2 + 6x+1| = 8
There are two solutions to an absolute value equation, one positive and one negative
x^2 +6x+1 =8 and x^2 +6x + 1 =-8
We have to solve both sets of equations
x^2 + 6x +1 = 8
Subtracting 8 from each side
x^2 +6x +1 -8 =0
x^2 +6x -7 =0
Factoring
What two numbers multiply to -7 and add to 6
-1 *7 = -7
-1 +7 = 6
These are the factors that we use ( -1 and 7)
( x-1) (x+7)=0
Using the zero product property
x-1 =0 x+7 =0
x=1 x=7
Now
x^2 +6x + 1 =-8
Add 8 to each side
x^2 +6x + 1+8 =0
x^2 + 6x+9 =0
Factoring
(x+3) (x+3) =0
Using the zero product property
x+3 =0 x+3 =0
x=-3
The solutions are
x=-3, x=1 , x=-7