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Solve for x (complex solution)
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\left(\sqrt{x-2017}\right)^{2}=x^{2}
Square both sides of the equation.
x-2017=x^{2}
Calculate \sqrt{x-2017} to the power of 2 and get x-2017.
x-2017-x^{2}=0
Subtract x^{2} from both sides.
-x^{2}+x-2017=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)\left(-2017\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and -2017 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-1\right)\left(-2017\right)}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4\left(-2017\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{1-8068}}{2\left(-1\right)}
Multiply 4 times -2017.
x=\frac{-1±\sqrt{-8067}}{2\left(-1\right)}
Add 1 to -8068.
x=\frac{-1±\sqrt{8067}i}{2\left(-1\right)}
Take the square root of -8067.
x=\frac{-1±\sqrt{8067}i}{-2}
Multiply 2 times -1.
x=\frac{-1+\sqrt{8067}i}{-2}
Now solve the equation x=\frac{-1±\sqrt{8067}i}{-2} when ± is plus. Add -1 to i\sqrt{8067}.
x=\frac{-\sqrt{8067}i+1}{2}
Divide -1+i\sqrt{8067} by -2.
x=\frac{-\sqrt{8067}i-1}{-2}
Now solve the equation x=\frac{-1±\sqrt{8067}i}{-2} when ± is minus. Subtract i\sqrt{8067} from -1.
x=\frac{1+\sqrt{8067}i}{2}
Divide -1-i\sqrt{8067} by -2.
x=\frac{-\sqrt{8067}i+1}{2} x=\frac{1+\sqrt{8067}i}{2}
The equation is now solved.
\sqrt{\frac{-\sqrt{8067}i+1}{2}-2017}=\frac{-\sqrt{8067}i+1}{2}
Substitute \frac{-\sqrt{8067}i+1}{2} for x in the equation \sqrt{x-2017}=x.
-\left(\frac{1}{2}-\frac{1}{2}i\times 8067^{\frac{1}{2}}\right)=-\frac{1}{2}i\times 8067^{\frac{1}{2}}+\frac{1}{2}
Simplify. The value x=\frac{-\sqrt{8067}i+1}{2} does not satisfy the equation.
\sqrt{\frac{1+\sqrt{8067}i}{2}-2017}=\frac{1+\sqrt{8067}i}{2}
Substitute \frac{1+\sqrt{8067}i}{2} for x in the equation \sqrt{x-2017}=x.
\frac{1}{2}+\frac{1}{2}i\times 8067^{\frac{1}{2}}=\frac{1}{2}+\frac{1}{2}i\times 8067^{\frac{1}{2}}
Simplify. The value x=\frac{1+\sqrt{8067}i}{2} satisfies the equation.
x=\frac{1+\sqrt{8067}i}{2}
Equation \sqrt{x-2017}=x has a unique solution.