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\sqrt{3-x}=-\left(-x+3\right)
Subtract -x+3 from both sides of the equation.
\sqrt{3-x}=-\left(-x\right)-3
To find the opposite of -x+3, find the opposite of each term.
\sqrt{3-x}=x-3
The opposite of -x is x.
\left(\sqrt{3-x}\right)^{2}=\left(x-3\right)^{2}
Square both sides of the equation.
3-x=\left(x-3\right)^{2}
Calculate \sqrt{3-x} to the power of 2 and get 3-x.
3-x=x^{2}-6x+9
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
3-x-x^{2}=-6x+9
Subtract x^{2} from both sides.
3-x-x^{2}+6x=9
Add 6x to both sides.
3+5x-x^{2}=9
Combine -x and 6x to get 5x.
3+5x-x^{2}-9=0
Subtract 9 from both sides.
-6+5x-x^{2}=0
Subtract 9 from 3 to get -6.
-x^{2}+5x-6=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=5 ab=-\left(-6\right)=6
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.
1,6 2,3
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 6.
1+6=7 2+3=5
Calculate the sum for each pair.
a=3 b=2
The solution is the pair that gives sum 5.
\left(-x^{2}+3x\right)+\left(2x-6\right)
Rewrite -x^{2}+5x-6 as \left(-x^{2}+3x\right)+\left(2x-6\right).
-x\left(x-3\right)+2\left(x-3\right)
Factor out -x in the first and 2 in the second group.
\left(x-3\right)\left(-x+2\right)
Factor out common term x-3 by using distributive property.
x=3 x=2
To find equation solutions, solve x-3=0 and -x+2=0.
\sqrt{3-3}-3+3=0
Substitute 3 for x in the equation \sqrt{3-x}-x+3=0.
0=0
Simplify. The value x=3 satisfies the equation.
\sqrt{3-2}-2+3=0
Substitute 2 for x in the equation \sqrt{3-x}-x+3=0.
2=0
Simplify. The value x=2 does not satisfy the equation.
x=3
Equation \sqrt{3-x}=x-3 has a unique solution.