Solve for k (complex solution)
k=-\frac{x-2}{x^{2}-x+1}
x\neq \frac{1+\sqrt{3}i}{2}\text{ and }x\neq \frac{-\sqrt{3}i+1}{2}
Solve for k
k=-\frac{x-2}{x^{2}-x+1}
Solve for x (complex solution)
\left\{\begin{matrix}x=\frac{\sqrt{1+6k-3k^{2}}+k-1}{2k}\text{; }x=\frac{-\sqrt{1+6k-3k^{2}}+k-1}{2k}\text{, }&k\neq 0\\x=2\text{, }&k=0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=\frac{\sqrt{1+6k-3k^{2}}+k-1}{2k}\text{; }x=\frac{-\sqrt{1+6k-3k^{2}}+k-1}{2k}\text{, }&k\neq 0\text{ and }k\geq -\frac{2\sqrt{3}}{3}+1\text{ and }k\leq \frac{2\sqrt{3}}{3}+1\\x=2\text{, }&k=0\end{matrix}\right.
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kx^{2}-\left(kx-x\right)+k-2=0
Use the distributive property to multiply k-1 by x.
kx^{2}-kx+x+k-2=0
To find the opposite of kx-x, find the opposite of each term.
kx^{2}-kx+k-2=-x
Subtract x from both sides. Anything subtracted from zero gives its negation.
kx^{2}-kx+k=-x+2
Add 2 to both sides.
\left(x^{2}-x+1\right)k=-x+2
Combine all terms containing k.
\left(x^{2}-x+1\right)k=2-x
The equation is in standard form.
\frac{\left(x^{2}-x+1\right)k}{x^{2}-x+1}=\frac{2-x}{x^{2}-x+1}
Divide both sides by x^{2}-x+1.
k=\frac{2-x}{x^{2}-x+1}
Dividing by x^{2}-x+1 undoes the multiplication by x^{2}-x+1.
kx^{2}-\left(kx-x\right)+k-2=0
Use the distributive property to multiply k-1 by x.
kx^{2}-kx+x+k-2=0
To find the opposite of kx-x, find the opposite of each term.
kx^{2}-kx+k-2=-x
Subtract x from both sides. Anything subtracted from zero gives its negation.
kx^{2}-kx+k=-x+2
Add 2 to both sides.
\left(x^{2}-x+1\right)k=-x+2
Combine all terms containing k.
\left(x^{2}-x+1\right)k=2-x
The equation is in standard form.
\frac{\left(x^{2}-x+1\right)k}{x^{2}-x+1}=\frac{2-x}{x^{2}-x+1}
Divide both sides by x^{2}-x+1.
k=\frac{2-x}{x^{2}-x+1}
Dividing by x^{2}-x+1 undoes the multiplication by x^{2}-x+1.
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Limits
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