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Solve for x (complex solution)
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x^{2}-2x+7=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(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 7}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\times 7}}{2}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4-28}}{2}
Multiply -4 times 7.
x=\frac{-\left(-2\right)±\sqrt{-24}}{2}
Add 4 to -28.
x=\frac{-\left(-2\right)±2\sqrt{6}i}{2}
Take the square root of -24.
x=\frac{2±2\sqrt{6}i}{2}
The opposite of -2 is 2.
x=\frac{2+2\sqrt{6}i}{2}
Now solve the equation x=\frac{2±2\sqrt{6}i}{2} when ± is plus. Add 2 to 2i\sqrt{6}.
x=1+\sqrt{6}i
Divide 2+2i\sqrt{6} by 2.
x=\frac{-2\sqrt{6}i+2}{2}
Now solve the equation x=\frac{2±2\sqrt{6}i}{2} when ± is minus. Subtract 2i\sqrt{6} from 2.
x=-\sqrt{6}i+1
Divide 2-2i\sqrt{6} by 2.
x=1+\sqrt{6}i x=-\sqrt{6}i+1
The equation is now solved.
x^{2}-2x+7=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.
x^{2}-2x+7-7=-7
Subtract 7 from both sides of the equation.
x^{2}-2x=-7
Subtracting 7 from itself leaves 0.
x^{2}-2x+1=-7+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=-6
Add -7 to 1.
\left(x-1\right)^{2}=-6
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{-6}
Take the square root of both sides of the equation.
x-1=\sqrt{6}i x-1=-\sqrt{6}i
Simplify.
x=1+\sqrt{6}i x=-\sqrt{6}i+1
Add 1 to both sides of the equation.