Solve for f (complex solution)
\left\{\begin{matrix}f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&\left(\alpha =0\text{ or }\beta =0\right)\text{ and }x=0\end{matrix}\right.
Solve for f
\left\{\begin{matrix}f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&\left(\alpha =0\text{ or }\beta =0\right)\text{ and }x=0\end{matrix}\right.
Solve for x (complex solution)
x=\frac{-\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}
x=\frac{\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}
Solve for x
x=\frac{-\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}
x=\frac{\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}\text{, }\left(\alpha >0\text{ and }\beta <0\right)\text{ or }\left(\beta >0\text{ and }\alpha <0\right)\text{ or }f\geq 2\sqrt{\alpha \beta }-\alpha -\beta \text{ or }f\leq -2\sqrt{\alpha \beta }-\alpha -\beta \text{ or }\left(\beta \geq 0\text{ and }\alpha \leq 0\right)\text{ or }\left(\alpha \geq 0\text{ and }\beta \leq 0\right)
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fx=x^{2}-x\beta -\alpha x+\beta \alpha
Use the distributive property to multiply x-\alpha by x-\beta .
xf=x^{2}-x\alpha -x\beta +\alpha \beta
The equation is in standard form.
\frac{xf}{x}=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}
Divide both sides by x.
f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}
Dividing by x undoes the multiplication by x.
fx=x^{2}-x\beta -\alpha x+\beta \alpha
Use the distributive property to multiply x-\alpha by x-\beta .
xf=x^{2}-x\alpha -x\beta +\alpha \beta
The equation is in standard form.
\frac{xf}{x}=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}
Divide both sides by x.
f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}
Dividing by x undoes the multiplication by x.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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Matrix
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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
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Limits
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