Solve for x (complex solution)
x=2+\sqrt{3}i\approx 2+1.732050808i
x=-\sqrt{3}i+2\approx 2-1.732050808i
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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.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}