Solve for x (complex solution)
x=2-i
x=2+i
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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.
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}