Skip to main content
Solve for x (complex solution)
Tick mark Image
Graph

Share

-85x^{2}+x=\frac{78}{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.
-85x^{2}+x-\frac{78}{5}=\frac{78}{5}-\frac{78}{5}
Subtract \frac{78}{5} from both sides of the equation.
-85x^{2}+x-\frac{78}{5}=0
Subtracting \frac{78}{5} from itself leaves 0.
x=\frac{-1±\sqrt{1^{2}-4\left(-85\right)\left(-\frac{78}{5}\right)}}{2\left(-85\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -85 for a, 1 for b, and -\frac{78}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-85\right)\left(-\frac{78}{5}\right)}}{2\left(-85\right)}
Square 1.
x=\frac{-1±\sqrt{1+340\left(-\frac{78}{5}\right)}}{2\left(-85\right)}
Multiply -4 times -85.
x=\frac{-1±\sqrt{1-5304}}{2\left(-85\right)}
Multiply 340 times -\frac{78}{5}.
x=\frac{-1±\sqrt{-5303}}{2\left(-85\right)}
Add 1 to -5304.
x=\frac{-1±\sqrt{5303}i}{2\left(-85\right)}
Take the square root of -5303.
x=\frac{-1±\sqrt{5303}i}{-170}
Multiply 2 times -85.
x=\frac{-1+\sqrt{5303}i}{-170}
Now solve the equation x=\frac{-1±\sqrt{5303}i}{-170} when ± is plus. Add -1 to i\sqrt{5303}.
x=\frac{-\sqrt{5303}i+1}{170}
Divide -1+i\sqrt{5303} by -170.
x=\frac{-\sqrt{5303}i-1}{-170}
Now solve the equation x=\frac{-1±\sqrt{5303}i}{-170} when ± is minus. Subtract i\sqrt{5303} from -1.
x=\frac{1+\sqrt{5303}i}{170}
Divide -1-i\sqrt{5303} by -170.
x=\frac{-\sqrt{5303}i+1}{170} x=\frac{1+\sqrt{5303}i}{170}
The equation is now solved.
-85x^{2}+x=\frac{78}{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{-85x^{2}+x}{-85}=\frac{\frac{78}{5}}{-85}
Divide both sides by -85.
x^{2}+\frac{1}{-85}x=\frac{\frac{78}{5}}{-85}
Dividing by -85 undoes the multiplication by -85.
x^{2}-\frac{1}{85}x=\frac{\frac{78}{5}}{-85}
Divide 1 by -85.
x^{2}-\frac{1}{85}x=-\frac{78}{425}
Divide \frac{78}{5} by -85.
x^{2}-\frac{1}{85}x+\left(-\frac{1}{170}\right)^{2}=-\frac{78}{425}+\left(-\frac{1}{170}\right)^{2}
Divide -\frac{1}{85}, the coefficient of the x term, by 2 to get -\frac{1}{170}. Then add the square of -\frac{1}{170} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{85}x+\frac{1}{28900}=-\frac{78}{425}+\frac{1}{28900}
Square -\frac{1}{170} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{85}x+\frac{1}{28900}=-\frac{5303}{28900}
Add -\frac{78}{425} to \frac{1}{28900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{170}\right)^{2}=-\frac{5303}{28900}
Factor x^{2}-\frac{1}{85}x+\frac{1}{28900}. 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{1}{170}\right)^{2}}=\sqrt{-\frac{5303}{28900}}
Take the square root of both sides of the equation.
x-\frac{1}{170}=\frac{\sqrt{5303}i}{170} x-\frac{1}{170}=-\frac{\sqrt{5303}i}{170}
Simplify.
x=\frac{1+\sqrt{5303}i}{170} x=\frac{-\sqrt{5303}i+1}{170}
Add \frac{1}{170} to both sides of the equation.