Solve for x (complex solution)
x=\frac{5+\sqrt{23}i}{2}\approx 2.5+2.397915762i
x=\frac{-\sqrt{23}i+5}{2}\approx 2.5-2.397915762i
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x^{2}-5x+6=-6
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^{2}-5x+6-\left(-6\right)=-6-\left(-6\right)
Add 6 to both sides of the equation.
x^{2}-5x+6-\left(-6\right)=0
Subtracting -6 from itself leaves 0.
x^{2}-5x+12=0
Subtract -6 from 6.
x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 12}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-5\right)±\sqrt{25-4\times 12}}{2}
Square -5.
x=\frac{-\left(-5\right)±\sqrt{25-48}}{2}
Multiply -4 times 12.
x=\frac{-\left(-5\right)±\sqrt{-23}}{2}
Add 25 to -48.
x=\frac{-\left(-5\right)±\sqrt{23}i}{2}
Take the square root of -23.
x=\frac{5±\sqrt{23}i}{2}
The opposite of -5 is 5.
x=\frac{5+\sqrt{23}i}{2}
Now solve the equation x=\frac{5±\sqrt{23}i}{2} when ± is plus. Add 5 to i\sqrt{23}.
x=\frac{-\sqrt{23}i+5}{2}
Now solve the equation x=\frac{5±\sqrt{23}i}{2} when ± is minus. Subtract i\sqrt{23} from 5.
x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}
The equation is now solved.
x^{2}-5x+6=-6
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}-5x+6-6=-6-6
Subtract 6 from both sides of the equation.
x^{2}-5x=-6-6
Subtracting 6 from itself leaves 0.
x^{2}-5x=-12
Subtract 6 from -6.
x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-12+\left(-\frac{5}{2}\right)^{2}
Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-5x+\frac{25}{4}=-12+\frac{25}{4}
Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-5x+\frac{25}{4}=-\frac{23}{4}
Add -12 to \frac{25}{4}.
\left(x-\frac{5}{2}\right)^{2}=-\frac{23}{4}
Factor x^{2}-5x+\frac{25}{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-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}
Take the square root of both sides of the equation.
x-\frac{5}{2}=\frac{\sqrt{23}i}{2} x-\frac{5}{2}=-\frac{\sqrt{23}i}{2}
Simplify.
x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}
Add \frac{5}{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}