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Solve for x (complex solution)
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-x^{2}+4x-5=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{-4±\sqrt{4^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}
Square 4.
x=\frac{-4±\sqrt{16+4\left(-5\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-4±\sqrt{16-20}}{2\left(-1\right)}
Multiply 4 times -5.
x=\frac{-4±\sqrt{-4}}{2\left(-1\right)}
Add 16 to -20.
x=\frac{-4±2i}{2\left(-1\right)}
Take the square root of -4.
x=\frac{-4±2i}{-2}
Multiply 2 times -1.
x=\frac{-4+2i}{-2}
Now solve the equation x=\frac{-4±2i}{-2} when ± is plus. Add -4 to 2i.
x=2-i
Divide -4+2i by -2.
x=\frac{-4-2i}{-2}
Now solve the equation x=\frac{-4±2i}{-2} when ± is minus. Subtract 2i from -4.
x=2+i
Divide -4-2i by -2.
x=2-i x=2+i
The equation is now solved.
-x^{2}+4x-5=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}+4x-5-\left(-5\right)=-\left(-5\right)
Add 5 to both sides of the equation.
-x^{2}+4x=-\left(-5\right)
Subtracting -5 from itself leaves 0.
-x^{2}+4x=5
Subtract -5 from 0.
\frac{-x^{2}+4x}{-1}=\frac{5}{-1}
Divide both sides by -1.
x^{2}+\frac{4}{-1}x=\frac{5}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-4x=\frac{5}{-1}
Divide 4 by -1.
x^{2}-4x=-5
Divide 5 by -1.
x^{2}-4x+\left(-2\right)^{2}=-5+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=-5+4
Square -2.
x^{2}-4x+4=-1
Add -5 to 4.
\left(x-2\right)^{2}=-1
Factor x^{2}-4x+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-2\right)^{2}}=\sqrt{-1}
Take the square root of both sides of the equation.
x-2=i x-2=-i
Simplify.
x=2+i x=2-i
Add 2 to both sides of the equation.