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Solve for x (complex solution)
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-x^{2}+18x=200
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^{2}+18x-200=200-200
Subtract 200 from both sides of the equation.
-x^{2}+18x-200=0
Subtracting 200 from itself leaves 0.
x=\frac{-18±\sqrt{18^{2}-4\left(-1\right)\left(-200\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 18 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-18±\sqrt{324-4\left(-1\right)\left(-200\right)}}{2\left(-1\right)}
Square 18.
x=\frac{-18±\sqrt{324+4\left(-200\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-18±\sqrt{324-800}}{2\left(-1\right)}
Multiply 4 times -200.
x=\frac{-18±\sqrt{-476}}{2\left(-1\right)}
Add 324 to -800.
x=\frac{-18±2\sqrt{119}i}{2\left(-1\right)}
Take the square root of -476.
x=\frac{-18±2\sqrt{119}i}{-2}
Multiply 2 times -1.
x=\frac{-18+2\sqrt{119}i}{-2}
Now solve the equation x=\frac{-18±2\sqrt{119}i}{-2} when ± is plus. Add -18 to 2i\sqrt{119}.
x=-\sqrt{119}i+9
Divide -18+2i\sqrt{119} by -2.
x=\frac{-2\sqrt{119}i-18}{-2}
Now solve the equation x=\frac{-18±2\sqrt{119}i}{-2} when ± is minus. Subtract 2i\sqrt{119} from -18.
x=9+\sqrt{119}i
Divide -18-2i\sqrt{119} by -2.
x=-\sqrt{119}i+9 x=9+\sqrt{119}i
The equation is now solved.
-x^{2}+18x=200
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{-x^{2}+18x}{-1}=\frac{200}{-1}
Divide both sides by -1.
x^{2}+\frac{18}{-1}x=\frac{200}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-18x=\frac{200}{-1}
Divide 18 by -1.
x^{2}-18x=-200
Divide 200 by -1.
x^{2}-18x+\left(-9\right)^{2}=-200+\left(-9\right)^{2}
Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-18x+81=-200+81
Square -9.
x^{2}-18x+81=-119
Add -200 to 81.
\left(x-9\right)^{2}=-119
Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-9\right)^{2}}=\sqrt{-119}
Take the square root of both sides of the equation.
x-9=\sqrt{119}i x-9=-\sqrt{119}i
Simplify.
x=9+\sqrt{119}i x=-\sqrt{119}i+9
Add 9 to both sides of the equation.