Solve for x (complex solution)
x=\frac{1+\sqrt{23}i}{2}\approx 0.5+2.397915762i
x=\frac{-\sqrt{23}i+1}{2}\approx 0.5-2.397915762i
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Quadratic Equation
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- \frac { 2 x ^ { 2 } - 2 x + 12 } { 4 - x ^ { 2 } } = 0
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-\left(2x^{2}-2x+12\right)=0
Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(-x-2\right).
-2x^{2}+2x-12=0
To find the opposite of 2x^{2}-2x+12, find the opposite of each term.
x=\frac{-2±\sqrt{2^{2}-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}
Square 2.
x=\frac{-2±\sqrt{4+8\left(-12\right)}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-2±\sqrt{4-96}}{2\left(-2\right)}
Multiply 8 times -12.
x=\frac{-2±\sqrt{-92}}{2\left(-2\right)}
Add 4 to -96.
x=\frac{-2±2\sqrt{23}i}{2\left(-2\right)}
Take the square root of -92.
x=\frac{-2±2\sqrt{23}i}{-4}
Multiply 2 times -2.
x=\frac{-2+2\sqrt{23}i}{-4}
Now solve the equation x=\frac{-2±2\sqrt{23}i}{-4} when ± is plus. Add -2 to 2i\sqrt{23}.
x=\frac{-\sqrt{23}i+1}{2}
Divide -2+2i\sqrt{23} by -4.
x=\frac{-2\sqrt{23}i-2}{-4}
Now solve the equation x=\frac{-2±2\sqrt{23}i}{-4} when ± is minus. Subtract 2i\sqrt{23} from -2.
x=\frac{1+\sqrt{23}i}{2}
Divide -2-2i\sqrt{23} by -4.
x=\frac{-\sqrt{23}i+1}{2} x=\frac{1+\sqrt{23}i}{2}
The equation is now solved.
-\left(2x^{2}-2x+12\right)=0
Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(-x-2\right).
-2x^{2}+2x-12=0
To find the opposite of 2x^{2}-2x+12, find the opposite of each term.
-2x^{2}+2x=12
Add 12 to both sides. Anything plus zero gives itself.
\frac{-2x^{2}+2x}{-2}=\frac{12}{-2}
Divide both sides by -2.
x^{2}+\frac{2}{-2}x=\frac{12}{-2}
Dividing by -2 undoes the multiplication by -2.
x^{2}-x=\frac{12}{-2}
Divide 2 by -2.
x^{2}-x=-6
Divide 12 by -2.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-6+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=-6+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=-\frac{23}{4}
Add -6 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=-\frac{23}{4}
Factor x^{2}-x+\frac{1}{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{1}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{23}i}{2} x-\frac{1}{2}=-\frac{\sqrt{23}i}{2}
Simplify.
x=\frac{1+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+1}{2}
Add \frac{1}{2} to both sides of the equation.
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