Solve for x (complex solution)
x=\frac{1+\sqrt{8067}i}{2}\approx 0.5+44.908239778i
Graph
Share
Copied to clipboard
\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.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}