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\sqrt{x}=1-x
Subtract x from both sides of the equation.
\left(\sqrt{x}\right)^{2}=\left(1-x\right)^{2}
Square both sides of the equation.
x=\left(1-x\right)^{2}
Calculate \sqrt{x} to the power of 2 and get x.
x=1-2x+x^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.
x-1=-2x+x^{2}
Subtract 1 from both sides.
x-1+2x=x^{2}
Add 2x to both sides.
3x-1=x^{2}
Combine x and 2x to get 3x.
3x-1-x^{2}=0
Subtract x^{2} from both sides.
-x^{2}+3x-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{-3±\sqrt{3^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}
Square 3.
x=\frac{-3±\sqrt{9+4\left(-1\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-3±\sqrt{9-4}}{2\left(-1\right)}
Multiply 4 times -1.
x=\frac{-3±\sqrt{5}}{2\left(-1\right)}
Add 9 to -4.
x=\frac{-3±\sqrt{5}}{-2}
Multiply 2 times -1.
x=\frac{\sqrt{5}-3}{-2}
Now solve the equation x=\frac{-3±\sqrt{5}}{-2} when ± is plus. Add -3 to \sqrt{5}.
x=\frac{3-\sqrt{5}}{2}
Divide -3+\sqrt{5} by -2.
x=\frac{-\sqrt{5}-3}{-2}
Now solve the equation x=\frac{-3±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from -3.
x=\frac{\sqrt{5}+3}{2}
Divide -3-\sqrt{5} by -2.
x=\frac{3-\sqrt{5}}{2} x=\frac{\sqrt{5}+3}{2}
The equation is now solved.
\frac{3-\sqrt{5}}{2}+\sqrt{\frac{3-\sqrt{5}}{2}}=1
Substitute \frac{3-\sqrt{5}}{2} for x in the equation x+\sqrt{x}=1.
1=1
Simplify. The value x=\frac{3-\sqrt{5}}{2} satisfies the equation.
\frac{\sqrt{5}+3}{2}+\sqrt{\frac{\sqrt{5}+3}{2}}=1
Substitute \frac{\sqrt{5}+3}{2} for x in the equation x+\sqrt{x}=1.
5^{\frac{1}{2}}+2=1
Simplify. The value x=\frac{\sqrt{5}+3}{2} does not satisfy the equation.
x=\frac{3-\sqrt{5}}{2}
Equation \sqrt{x}=1-x has a unique solution.