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\left(\sqrt{2-x}\right)^{2}=\left(x-1\right)^{2}
Square both sides of the equation.
2-x=\left(x-1\right)^{2}
Calculate \sqrt{2-x} to the power of 2 and get 2-x.
2-x=x^{2}-2x+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
2-x-x^{2}=-2x+1
Subtract x^{2} from both sides.
2-x-x^{2}+2x=1
Add 2x to both sides.
2+x-x^{2}=1
Combine -x and 2x to get x.
2+x-x^{2}-1=0
Subtract 1 from both sides.
1+x-x^{2}=0
Subtract 1 from 2 to get 1.
-x^{2}+x+1=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-1\right)}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{5}}{2\left(-1\right)}
Add 1 to 4.
x=\frac{-1±\sqrt{5}}{-2}
Multiply 2 times -1.
x=\frac{\sqrt{5}-1}{-2}
Now solve the equation x=\frac{-1±\sqrt{5}}{-2} when ± is plus. Add -1 to \sqrt{5}.
x=\frac{1-\sqrt{5}}{2}
Divide -1+\sqrt{5} by -2.
x=\frac{-\sqrt{5}-1}{-2}
Now solve the equation x=\frac{-1±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from -1.
x=\frac{\sqrt{5}+1}{2}
Divide -1-\sqrt{5} by -2.
x=\frac{1-\sqrt{5}}{2} x=\frac{\sqrt{5}+1}{2}
The equation is now solved.
\sqrt{2-\frac{1-\sqrt{5}}{2}}=\frac{1-\sqrt{5}}{2}-1
Substitute \frac{1-\sqrt{5}}{2} for x in the equation \sqrt{2-x}=x-1.
\frac{1}{2}+\frac{1}{2}\times 5^{\frac{1}{2}}=-\frac{1}{2}-\frac{1}{2}\times 5^{\frac{1}{2}}
Simplify. The value x=\frac{1-\sqrt{5}}{2} does not satisfy the equation because the left and the right hand side have opposite signs.
\sqrt{2-\frac{\sqrt{5}+1}{2}}=\frac{\sqrt{5}+1}{2}-1
Substitute \frac{\sqrt{5}+1}{2} for x in the equation \sqrt{2-x}=x-1.
-\left(\frac{1}{2}-\frac{1}{2}\times 5^{\frac{1}{2}}\right)=\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}
Simplify. The value x=\frac{\sqrt{5}+1}{2} satisfies the equation.
x=\frac{\sqrt{5}+1}{2}
Equation \sqrt{2-x}=x-1 has a unique solution.