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110x^{2}-110x+200=0
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=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 110\times 200}}{2\times 110}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 110 for a, -110 for b, and 200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-110\right)±\sqrt{12100-4\times 110\times 200}}{2\times 110}
Square -110.
x=\frac{-\left(-110\right)±\sqrt{12100-440\times 200}}{2\times 110}
Multiply -4 times 110.
x=\frac{-\left(-110\right)±\sqrt{12100-88000}}{2\times 110}
Multiply -440 times 200.
x=\frac{-\left(-110\right)±\sqrt{-75900}}{2\times 110}
Add 12100 to -88000.
x=\frac{-\left(-110\right)±10\sqrt{759}i}{2\times 110}
Take the square root of -75900.
x=\frac{110±10\sqrt{759}i}{2\times 110}
The opposite of -110 is 110.
x=\frac{110±10\sqrt{759}i}{220}
Multiply 2 times 110.
x=\frac{110+10\sqrt{759}i}{220}
Now solve the equation x=\frac{110±10\sqrt{759}i}{220} when ± is plus. Add 110 to 10i\sqrt{759}.
x=\frac{\sqrt{759}i}{22}+\frac{1}{2}
Divide 110+10i\sqrt{759} by 220.
x=\frac{-10\sqrt{759}i+110}{220}
Now solve the equation x=\frac{110±10\sqrt{759}i}{220} when ± is minus. Subtract 10i\sqrt{759} from 110.
x=-\frac{\sqrt{759}i}{22}+\frac{1}{2}
Divide 110-10i\sqrt{759} by 220.
x=\frac{\sqrt{759}i}{22}+\frac{1}{2} x=-\frac{\sqrt{759}i}{22}+\frac{1}{2}
The equation is now solved.
110x^{2}-110x+200=0
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.
110x^{2}-110x+200-200=-200
Subtract 200 from both sides of the equation.
110x^{2}-110x=-200
Subtracting 200 from itself leaves 0.
\frac{110x^{2}-110x}{110}=-\frac{200}{110}
Divide both sides by 110.
x^{2}+\left(-\frac{110}{110}\right)x=-\frac{200}{110}
Dividing by 110 undoes the multiplication by 110.
x^{2}-x=-\frac{200}{110}
Divide -110 by 110.
x^{2}-x=-\frac{20}{11}
Reduce the fraction \frac{-200}{110} to lowest terms by extracting and canceling out 10.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{20}{11}+\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}=-\frac{20}{11}+\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{69}{44}
Add -\frac{20}{11} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{2}\right)^{2}=-\frac{69}{44}
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{69}{44}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{759}i}{22} x-\frac{1}{2}=-\frac{\sqrt{759}i}{22}
Simplify.
x=\frac{\sqrt{759}i}{22}+\frac{1}{2} x=-\frac{\sqrt{759}i}{22}+\frac{1}{2}
Add \frac{1}{2} to both sides of the equation.