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Solve for x (complex solution)
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-x^{2}+3x=69
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^{2}+3x-69=69-69
Subtract 69 from both sides of the equation.
-x^{2}+3x-69=0
Subtracting 69 from itself leaves 0.
x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-69\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -69 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-69\right)}}{2\left(-1\right)}
Square 3.
x=\frac{-3±\sqrt{9+4\left(-69\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-3±\sqrt{9-276}}{2\left(-1\right)}
Multiply 4 times -69.
x=\frac{-3±\sqrt{-267}}{2\left(-1\right)}
Add 9 to -276.
x=\frac{-3±\sqrt{267}i}{2\left(-1\right)}
Take the square root of -267.
x=\frac{-3±\sqrt{267}i}{-2}
Multiply 2 times -1.
x=\frac{-3+\sqrt{267}i}{-2}
Now solve the equation x=\frac{-3±\sqrt{267}i}{-2} when ± is plus. Add -3 to i\sqrt{267}.
x=\frac{-\sqrt{267}i+3}{2}
Divide -3+i\sqrt{267} by -2.
x=\frac{-\sqrt{267}i-3}{-2}
Now solve the equation x=\frac{-3±\sqrt{267}i}{-2} when ± is minus. Subtract i\sqrt{267} from -3.
x=\frac{3+\sqrt{267}i}{2}
Divide -3-i\sqrt{267} by -2.
x=\frac{-\sqrt{267}i+3}{2} x=\frac{3+\sqrt{267}i}{2}
The equation is now solved.
-x^{2}+3x=69
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{-x^{2}+3x}{-1}=\frac{69}{-1}
Divide both sides by -1.
x^{2}+\frac{3}{-1}x=\frac{69}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-3x=\frac{69}{-1}
Divide 3 by -1.
x^{2}-3x=-69
Divide 69 by -1.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-69+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-3x+\frac{9}{4}=-69+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=-\frac{267}{4}
Add -69 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=-\frac{267}{4}
Factor x^{2}-3x+\frac{9}{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{3}{2}\right)^{2}}=\sqrt{-\frac{267}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{\sqrt{267}i}{2} x-\frac{3}{2}=-\frac{\sqrt{267}i}{2}
Simplify.
x=\frac{3+\sqrt{267}i}{2} x=\frac{-\sqrt{267}i+3}{2}
Add \frac{3}{2} to both sides of the equation.