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\sqrt{x+8}=-4+x
Subtract -x from both sides of the equation.
\left(\sqrt{x+8}\right)^{2}=\left(-4+x\right)^{2}
Square both sides of the equation.
x+8=\left(-4+x\right)^{2}
Calculate \sqrt{x+8} to the power of 2 and get x+8.
x+8=16-8x+x^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-4+x\right)^{2}.
x+8-16=-8x+x^{2}
Subtract 16 from both sides.
x-8=-8x+x^{2}
Subtract 16 from 8 to get -8.
x-8+8x=x^{2}
Add 8x to both sides.
9x-8=x^{2}
Combine x and 8x to get 9x.
9x-8-x^{2}=0
Subtract x^{2} from both sides.
-x^{2}+9x-8=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=9 ab=-\left(-8\right)=8
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-8. To find a and b, set up a system to be solved.
1,8 2,4
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 8.
1+8=9 2+4=6
Calculate the sum for each pair.
a=8 b=1
The solution is the pair that gives sum 9.
\left(-x^{2}+8x\right)+\left(x-8\right)
Rewrite -x^{2}+9x-8 as \left(-x^{2}+8x\right)+\left(x-8\right).
-x\left(x-8\right)+x-8
Factor out -x in -x^{2}+8x.
\left(x-8\right)\left(-x+1\right)
Factor out common term x-8 by using distributive property.
x=8 x=1
To find equation solutions, solve x-8=0 and -x+1=0.
\sqrt{8+8}-8=-4
Substitute 8 for x in the equation \sqrt{x+8}-x=-4.
-4=-4
Simplify. The value x=8 satisfies the equation.
\sqrt{1+8}-1=-4
Substitute 1 for x in the equation \sqrt{x+8}-x=-4.
2=-4
Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.
x=8
Equation \sqrt{x+8}=x-4 has a unique solution.