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Solve for x (complex solution)
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x^{2}-4x+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(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 7}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\times 7}}{2}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16-28}}{2}
Multiply -4 times 7.
x=\frac{-\left(-4\right)±\sqrt{-12}}{2}
Add 16 to -28.
x=\frac{-\left(-4\right)±2\sqrt{3}i}{2}
Take the square root of -12.
x=\frac{4±2\sqrt{3}i}{2}
The opposite of -4 is 4.
x=\frac{4+2\sqrt{3}i}{2}
Now solve the equation x=\frac{4±2\sqrt{3}i}{2} when ± is plus. Add 4 to 2i\sqrt{3}.
x=2+\sqrt{3}i
Divide 4+2i\sqrt{3} by 2.
x=\frac{-2\sqrt{3}i+4}{2}
Now solve the equation x=\frac{4±2\sqrt{3}i}{2} when ± is minus. Subtract 2i\sqrt{3} from 4.
x=-\sqrt{3}i+2
Divide 4-2i\sqrt{3} by 2.
x=2+\sqrt{3}i x=-\sqrt{3}i+2
The equation is now solved.
x^{2}-4x+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}-4x+7-7=-7
Subtract 7 from both sides of the equation.
x^{2}-4x=-7
Subtracting 7 from itself leaves 0.
x^{2}-4x+\left(-2\right)^{2}=-7+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=-7+4
Square -2.
x^{2}-4x+4=-3
Add -7 to 4.
\left(x-2\right)^{2}=-3
Factor x^{2}-4x+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-2\right)^{2}}=\sqrt{-3}
Take the square root of both sides of the equation.
x-2=\sqrt{3}i x-2=-\sqrt{3}i
Simplify.
x=2+\sqrt{3}i x=-\sqrt{3}i+2
Add 2 to both sides of the equation.