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-5x^{2}+10x=\frac{9}{5}
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}+10x-\frac{9}{5}=\frac{9}{5}-\frac{9}{5}
Subtract \frac{9}{5} from both sides of the equation.
-5x^{2}+10x-\frac{9}{5}=0
Subtracting \frac{9}{5} from itself leaves 0.
x=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\left(-\frac{9}{5}\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 10 for b, and -\frac{9}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\left(-5\right)\left(-\frac{9}{5}\right)}}{2\left(-5\right)}
Square 10.
x=\frac{-10±\sqrt{100+20\left(-\frac{9}{5}\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-10±\sqrt{100-36}}{2\left(-5\right)}
Multiply 20 times -\frac{9}{5}.
x=\frac{-10±\sqrt{64}}{2\left(-5\right)}
Add 100 to -36.
x=\frac{-10±8}{2\left(-5\right)}
Take the square root of 64.
x=\frac{-10±8}{-10}
Multiply 2 times -5.
x=-\frac{2}{-10}
Now solve the equation x=\frac{-10±8}{-10} when ± is plus. Add -10 to 8.
x=\frac{1}{5}
Reduce the fraction \frac{-2}{-10} to lowest terms by extracting and canceling out 2.
x=-\frac{18}{-10}
Now solve the equation x=\frac{-10±8}{-10} when ± is minus. Subtract 8 from -10.
x=\frac{9}{5}
Reduce the fraction \frac{-18}{-10} to lowest terms by extracting and canceling out 2.
x=\frac{1}{5} x=\frac{9}{5}
The equation is now solved.
-5x^{2}+10x=\frac{9}{5}
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}+10x}{-5}=\frac{\frac{9}{5}}{-5}
Divide both sides by -5.
x^{2}+\frac{10}{-5}x=\frac{\frac{9}{5}}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}-2x=\frac{\frac{9}{5}}{-5}
Divide 10 by -5.
x^{2}-2x=-\frac{9}{25}
Divide \frac{9}{5} by -5.
x^{2}-2x+1=-\frac{9}{25}+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-2x+1=\frac{16}{25}
Add -\frac{9}{25} to 1.
\left(x-1\right)^{2}=\frac{16}{25}
Factor x^{2}-2x+1. 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-1\right)^{2}}=\sqrt{\frac{16}{25}}
Take the square root of both sides of the equation.
x-1=\frac{4}{5} x-1=-\frac{4}{5}
Simplify.
x=\frac{9}{5} x=\frac{1}{5}
Add 1 to both sides of the equation.