Solve for x (complex solution)
x=1+i
x=1-i
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3x^{2}-6x+8=2
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.
3x^{2}-6x+8-2=2-2
Subtract 2 from both sides of the equation.
3x^{2}-6x+8-2=0
Subtracting 2 from itself leaves 0.
3x^{2}-6x+6=0
Subtract 2 from 8.
x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\times 6}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\times 6}}{2\times 3}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36-12\times 6}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-6\right)±\sqrt{36-72}}{2\times 3}
Multiply -12 times 6.
x=\frac{-\left(-6\right)±\sqrt{-36}}{2\times 3}
Add 36 to -72.
x=\frac{-\left(-6\right)±6i}{2\times 3}
Take the square root of -36.
x=\frac{6±6i}{2\times 3}
The opposite of -6 is 6.
x=\frac{6±6i}{6}
Multiply 2 times 3.
x=\frac{6+6i}{6}
Now solve the equation x=\frac{6±6i}{6} when ± is plus. Add 6 to 6i.
x=1+i
Divide 6+6i by 6.
x=\frac{6-6i}{6}
Now solve the equation x=\frac{6±6i}{6} when ± is minus. Subtract 6i from 6.
x=1-i
Divide 6-6i by 6.
x=1+i x=1-i
The equation is now solved.
3x^{2}-6x+8=2
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.
3x^{2}-6x+8-8=2-8
Subtract 8 from both sides of the equation.
3x^{2}-6x=2-8
Subtracting 8 from itself leaves 0.
3x^{2}-6x=-6
Subtract 8 from 2.
\frac{3x^{2}-6x}{3}=-\frac{6}{3}
Divide both sides by 3.
x^{2}+\left(-\frac{6}{3}\right)x=-\frac{6}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-2x=-\frac{6}{3}
Divide -6 by 3.
x^{2}-2x=-2
Divide -6 by 3.
x^{2}-2x+1=-2+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=-1
Add -2 to 1.
\left(x-1\right)^{2}=-1
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{-1}
Take the square root of both sides of the equation.
x-1=i x-1=-i
Simplify.
x=1+i x=1-i
Add 1 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}