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Solve for x (complex solution)
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-5x^{2}+8x=9
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-9=9-9
Subtract 9 from both sides of the equation.
-5x^{2}+8x-9=0
Subtracting 9 from itself leaves 0.
x=\frac{-8±\sqrt{8^{2}-4\left(-5\right)\left(-9\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\left(-5\right)\left(-9\right)}}{2\left(-5\right)}
Square 8.
x=\frac{-8±\sqrt{64+20\left(-9\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-8±\sqrt{64-180}}{2\left(-5\right)}
Multiply 20 times -9.
x=\frac{-8±\sqrt{-116}}{2\left(-5\right)}
Add 64 to -180.
x=\frac{-8±2\sqrt{29}i}{2\left(-5\right)}
Take the square root of -116.
x=\frac{-8±2\sqrt{29}i}{-10}
Multiply 2 times -5.
x=\frac{-8+2\sqrt{29}i}{-10}
Now solve the equation x=\frac{-8±2\sqrt{29}i}{-10} when ± is plus. Add -8 to 2i\sqrt{29}.
x=\frac{-\sqrt{29}i+4}{5}
Divide -8+2i\sqrt{29} by -10.
x=\frac{-2\sqrt{29}i-8}{-10}
Now solve the equation x=\frac{-8±2\sqrt{29}i}{-10} when ± is minus. Subtract 2i\sqrt{29} from -8.
x=\frac{4+\sqrt{29}i}{5}
Divide -8-2i\sqrt{29} by -10.
x=\frac{-\sqrt{29}i+4}{5} x=\frac{4+\sqrt{29}i}{5}
The equation is now solved.
-5x^{2}+8x=9
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{9}{-5}
Divide both sides by -5.
x^{2}+\frac{8}{-5}x=\frac{9}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}-\frac{8}{5}x=\frac{9}{-5}
Divide 8 by -5.
x^{2}-\frac{8}{5}x=-\frac{9}{5}
Divide 9 by -5.
x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{9}{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{9}{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{29}{25}
Add -\frac{9}{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{29}{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{29}{25}}
Take the square root of both sides of the equation.
x-\frac{4}{5}=\frac{\sqrt{29}i}{5} x-\frac{4}{5}=-\frac{\sqrt{29}i}{5}
Simplify.
x=\frac{4+\sqrt{29}i}{5} x=\frac{-\sqrt{29}i+4}{5}
Add \frac{4}{5} to both sides of the equation.