Solve for k
k=\frac{x}{11}+\frac{103}{33}+\frac{6}{11x}
x\neq 0\text{ and }x\neq -3
Solve for x (complex solution)
\left\{\begin{matrix}\\x=\frac{\sqrt{1089k^{2}-6798k+10393}}{6}+\frac{11k}{2}-\frac{103}{6}\text{, }&\text{unconditionally}\\x=-\frac{\sqrt{1089k^{2}-6798k+10393}}{6}+\frac{11k}{2}-\frac{103}{6}\text{, }&k\neq \frac{8}{3}\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=-\frac{\sqrt{1089k^{2}-6798k+10393}}{6}+\frac{11k}{2}-\frac{103}{6}\text{, }&\left(k\leq -\frac{2\sqrt{6}}{11}+\frac{103}{33}\text{ and }k\neq \frac{8}{3}\right)\text{ or }k\geq \frac{2\sqrt{6}}{11}+\frac{103}{33}\\x=\frac{\sqrt{1089k^{2}-6798k+10393}}{6}+\frac{11k}{2}-\frac{103}{6}\text{, }&k\geq \frac{2\sqrt{6}}{11}+\frac{103}{33}\text{ or }k\leq -\frac{2\sqrt{6}}{11}+\frac{103}{33}\end{matrix}\right.
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\left(x+3\right)\left(3x+6\right)=11x\left(3k-8\right)
Multiply both sides of the equation by 11x\left(x+3\right), the least common multiple of 8x+3x,x+3.
3x^{2}+15x+18=11x\left(3k-8\right)
Use the distributive property to multiply x+3 by 3x+6 and combine like terms.
3x^{2}+15x+18=33kx-88x
Use the distributive property to multiply 11x by 3k-8.
33kx-88x=3x^{2}+15x+18
Swap sides so that all variable terms are on the left hand side.
33kx=3x^{2}+15x+18+88x
Add 88x to both sides.
33kx=3x^{2}+103x+18
Combine 15x and 88x to get 103x.
33xk=3x^{2}+103x+18
The equation is in standard form.
\frac{33xk}{33x}=\frac{3x^{2}+103x+18}{33x}
Divide both sides by 33x.
k=\frac{3x^{2}+103x+18}{33x}
Dividing by 33x undoes the multiplication by 33x.
k=\frac{x}{11}+\frac{103}{33}+\frac{6}{11x}
Divide 3x^{2}+103x+18 by 33x.
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