Solve for x (complex solution)
x=\frac{-\sqrt{22}i+5}{2}\approx 2.5-2.34520788i
x=\frac{5+\sqrt{22}i}{2}\approx 2.5+2.34520788i
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-4x^{2}+20x-47=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{-20±\sqrt{20^{2}-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 20 for b, and -47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-20±\sqrt{400-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}
Square 20.
x=\frac{-20±\sqrt{400+16\left(-47\right)}}{2\left(-4\right)}
Multiply -4 times -4.
x=\frac{-20±\sqrt{400-752}}{2\left(-4\right)}
Multiply 16 times -47.
x=\frac{-20±\sqrt{-352}}{2\left(-4\right)}
Add 400 to -752.
x=\frac{-20±4\sqrt{22}i}{2\left(-4\right)}
Take the square root of -352.
x=\frac{-20±4\sqrt{22}i}{-8}
Multiply 2 times -4.
x=\frac{-20+4\sqrt{22}i}{-8}
Now solve the equation x=\frac{-20±4\sqrt{22}i}{-8} when ± is plus. Add -20 to 4i\sqrt{22}.
x=\frac{-\sqrt{22}i+5}{2}
Divide -20+4i\sqrt{22} by -8.
x=\frac{-4\sqrt{22}i-20}{-8}
Now solve the equation x=\frac{-20±4\sqrt{22}i}{-8} when ± is minus. Subtract 4i\sqrt{22} from -20.
x=\frac{5+\sqrt{22}i}{2}
Divide -20-4i\sqrt{22} by -8.
x=\frac{-\sqrt{22}i+5}{2} x=\frac{5+\sqrt{22}i}{2}
The equation is now solved.
-4x^{2}+20x-47=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.
-4x^{2}+20x-47-\left(-47\right)=-\left(-47\right)
Add 47 to both sides of the equation.
-4x^{2}+20x=-\left(-47\right)
Subtracting -47 from itself leaves 0.
-4x^{2}+20x=47
Subtract -47 from 0.
\frac{-4x^{2}+20x}{-4}=\frac{47}{-4}
Divide both sides by -4.
x^{2}+\frac{20}{-4}x=\frac{47}{-4}
Dividing by -4 undoes the multiplication by -4.
x^{2}-5x=\frac{47}{-4}
Divide 20 by -4.
x^{2}-5x=-\frac{47}{4}
Divide 47 by -4.
x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{47}{4}+\left(-\frac{5}{2}\right)^{2}
Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-5x+\frac{25}{4}=\frac{-47+25}{4}
Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-5x+\frac{25}{4}=-\frac{11}{2}
Add -\frac{47}{4} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{5}{2}\right)^{2}=-\frac{11}{2}
Factor x^{2}-5x+\frac{25}{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{5}{2}\right)^{2}}=\sqrt{-\frac{11}{2}}
Take the square root of both sides of the equation.
x-\frac{5}{2}=\frac{\sqrt{22}i}{2} x-\frac{5}{2}=-\frac{\sqrt{22}i}{2}
Simplify.
x=\frac{5+\sqrt{22}i}{2} x=\frac{-\sqrt{22}i+5}{2}
Add \frac{5}{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}