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Solve for x (complex solution)
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x^{2}-6x+14=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(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 14}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\times 14}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36-56}}{2}
Multiply -4 times 14.
x=\frac{-\left(-6\right)±\sqrt{-20}}{2}
Add 36 to -56.
x=\frac{-\left(-6\right)±2\sqrt{5}i}{2}
Take the square root of -20.
x=\frac{6±2\sqrt{5}i}{2}
The opposite of -6 is 6.
x=\frac{6+2\sqrt{5}i}{2}
Now solve the equation x=\frac{6±2\sqrt{5}i}{2} when ± is plus. Add 6 to 2i\sqrt{5}.
x=3+\sqrt{5}i
Divide 6+2i\sqrt{5} by 2.
x=\frac{-2\sqrt{5}i+6}{2}
Now solve the equation x=\frac{6±2\sqrt{5}i}{2} when ± is minus. Subtract 2i\sqrt{5} from 6.
x=-\sqrt{5}i+3
Divide 6-2i\sqrt{5} by 2.
x=3+\sqrt{5}i x=-\sqrt{5}i+3
The equation is now solved.
x^{2}-6x+14=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}-6x+14-14=-14
Subtract 14 from both sides of the equation.
x^{2}-6x=-14
Subtracting 14 from itself leaves 0.
x^{2}-6x+\left(-3\right)^{2}=-14+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=-14+9
Square -3.
x^{2}-6x+9=-5
Add -14 to 9.
\left(x-3\right)^{2}=-5
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{-5}
Take the square root of both sides of the equation.
x-3=\sqrt{5}i x-3=-\sqrt{5}i
Simplify.
x=3+\sqrt{5}i x=-\sqrt{5}i+3
Add 3 to both sides of the equation.