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Solve for x (complex solution)
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-5x^{2}-8x=4
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.
-5x^{2}-8x-4=4-4
Subtract 4 from both sides of the equation.
-5x^{2}-8x-4=0
Subtracting 4 from itself leaves 0.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-5\right)\left(-4\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -8 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\left(-5\right)\left(-4\right)}}{2\left(-5\right)}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64+20\left(-4\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-\left(-8\right)±\sqrt{64-80}}{2\left(-5\right)}
Multiply 20 times -4.
x=\frac{-\left(-8\right)±\sqrt{-16}}{2\left(-5\right)}
Add 64 to -80.
x=\frac{-\left(-8\right)±4i}{2\left(-5\right)}
Take the square root of -16.
x=\frac{8±4i}{2\left(-5\right)}
The opposite of -8 is 8.
x=\frac{8±4i}{-10}
Multiply 2 times -5.
x=\frac{8+4i}{-10}
Now solve the equation x=\frac{8±4i}{-10} when ± is plus. Add 8 to 4i.
x=-\frac{4}{5}-\frac{2}{5}i
Divide 8+4i by -10.
x=\frac{8-4i}{-10}
Now solve the equation x=\frac{8±4i}{-10} when ± is minus. Subtract 4i from 8.
x=-\frac{4}{5}+\frac{2}{5}i
Divide 8-4i by -10.
x=-\frac{4}{5}-\frac{2}{5}i x=-\frac{4}{5}+\frac{2}{5}i
The equation is now solved.
-5x^{2}-8x=4
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{-5x^{2}-8x}{-5}=\frac{4}{-5}
Divide both sides by -5.
x^{2}+\left(-\frac{8}{-5}\right)x=\frac{4}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}+\frac{8}{5}x=\frac{4}{-5}
Divide -8 by -5.
x^{2}+\frac{8}{5}x=-\frac{4}{5}
Divide 4 by -5.
x^{2}+\frac{8}{5}x+\left(\frac{4}{5}\right)^{2}=-\frac{4}{5}+\left(\frac{4}{5}\right)^{2}
Divide \frac{8}{5}, the coefficient of the x term, by 2 to get \frac{4}{5}. Then add the square of \frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{8}{5}x+\frac{16}{25}=-\frac{4}{5}+\frac{16}{25}
Square \frac{4}{5} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{8}{5}x+\frac{16}{25}=-\frac{4}{25}
Add -\frac{4}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{4}{5}\right)^{2}=-\frac{4}{25}
Factor x^{2}+\frac{8}{5}x+\frac{16}{25}. 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{4}{5}\right)^{2}}=\sqrt{-\frac{4}{25}}
Take the square root of both sides of the equation.
x+\frac{4}{5}=\frac{2}{5}i x+\frac{4}{5}=-\frac{2}{5}i
Simplify.
x=-\frac{4}{5}+\frac{2}{5}i x=-\frac{4}{5}-\frac{2}{5}i
Subtract \frac{4}{5} from both sides of the equation.