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\left(\sqrt{x}\right)^{2}=\left(x-6\right)^{2}
Square both sides of the equation.
x=\left(x-6\right)^{2}
Calculate \sqrt{x} to the power of 2 and get x.
x=x^{2}-12x+36
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.
x-x^{2}=-12x+36
Subtract x^{2} from both sides.
x-x^{2}+12x=36
Add 12x to both sides.
13x-x^{2}=36
Combine x and 12x to get 13x.
13x-x^{2}-36=0
Subtract 36 from both sides.
-x^{2}+13x-36=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=13 ab=-\left(-36\right)=36
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-36. To find a and b, set up a system to be solved.
1,36 2,18 3,12 4,9 6,6
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 36.
1+36=37 2+18=20 3+12=15 4+9=13 6+6=12
Calculate the sum for each pair.
a=9 b=4
The solution is the pair that gives sum 13.
\left(-x^{2}+9x\right)+\left(4x-36\right)
Rewrite -x^{2}+13x-36 as \left(-x^{2}+9x\right)+\left(4x-36\right).
-x\left(x-9\right)+4\left(x-9\right)
Factor out -x in the first and 4 in the second group.
\left(x-9\right)\left(-x+4\right)
Factor out common term x-9 by using distributive property.
x=9 x=4
To find equation solutions, solve x-9=0 and -x+4=0.
\sqrt{9}=9-6
Substitute 9 for x in the equation \sqrt{x}=x-6.
3=3
Simplify. The value x=9 satisfies the equation.
\sqrt{4}=4-6
Substitute 4 for x in the equation \sqrt{x}=x-6.
2=-2
Simplify. The value x=4 does not satisfy the equation because the left and the right hand side have opposite signs.
x=9
Equation \sqrt{x}=x-6 has a unique solution.