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Solve for x (complex solution)
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-x^{2}+x-45=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(-45\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-1\right)\left(-45\right)}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4\left(-45\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{1-180}}{2\left(-1\right)}
Multiply 4 times -45.
x=\frac{-1±\sqrt{-179}}{2\left(-1\right)}
Add 1 to -180.
x=\frac{-1±\sqrt{179}i}{2\left(-1\right)}
Take the square root of -179.
x=\frac{-1±\sqrt{179}i}{-2}
Multiply 2 times -1.
x=\frac{-1+\sqrt{179}i}{-2}
Now solve the equation x=\frac{-1±\sqrt{179}i}{-2} when ± is plus. Add -1 to i\sqrt{179}.
x=\frac{-\sqrt{179}i+1}{2}
Divide -1+i\sqrt{179} by -2.
x=\frac{-\sqrt{179}i-1}{-2}
Now solve the equation x=\frac{-1±\sqrt{179}i}{-2} when ± is minus. Subtract i\sqrt{179} from -1.
x=\frac{1+\sqrt{179}i}{2}
Divide -1-i\sqrt{179} by -2.
x=\frac{-\sqrt{179}i+1}{2} x=\frac{1+\sqrt{179}i}{2}
The equation is now solved.
-x^{2}+x-45=0
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.
-x^{2}+x-45-\left(-45\right)=-\left(-45\right)
Add 45 to both sides of the equation.
-x^{2}+x=-\left(-45\right)
Subtracting -45 from itself leaves 0.
-x^{2}+x=45
Subtract -45 from 0.
\frac{-x^{2}+x}{-1}=\frac{45}{-1}
Divide both sides by -1.
x^{2}+\frac{1}{-1}x=\frac{45}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-x=\frac{45}{-1}
Divide 1 by -1.
x^{2}-x=-45
Divide 45 by -1.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-45+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=-45+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=-\frac{179}{4}
Add -45 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=-\frac{179}{4}
Factor x^{2}-x+\frac{1}{4}. 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-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{179}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{179}i}{2} x-\frac{1}{2}=-\frac{\sqrt{179}i}{2}
Simplify.
x=\frac{1+\sqrt{179}i}{2} x=\frac{-\sqrt{179}i+1}{2}
Add \frac{1}{2} to both sides of the equation.