Solve kx^2+3kx+x+5=0

$$k x ^ { 2 } + 3 k x + x + 5 = 0$$

Solve for k

$$k=-\frac{x+5}{x\left(x+3\right)}$$

$$x\neq -3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Subtract x from both sides. Anything subtracted from zero gives its negation.

$$kx^{2}+3kx+5=-x$$

Subtract 5 from both sides.

$$kx^{2}+3kx=-x-5$$

Combine all terms containing k.

$$\left(x^{2}+3x\right)k=-x-5$$

Divide both sides by x^{2}+3x.

$$\frac{\left(x^{2}+3x\right)k}{x^{2}+3x}=\frac{-x-5}{x^{2}+3x}$$

Dividing by x^{2}+3x undoes the multiplication by x^{2}+3x.

$$k=\frac{-x-5}{x^{2}+3x}$$

Divide -x-5 by x^{2}+3x.

$$k=-\frac{x+5}{x\left(x+3\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{9k^{2}-14k+1}-3k-1}{2k}\text{; }x=-\frac{\sqrt{9k^{2}-14k+1}+3k+1}{2k}\text{, }&k\neq 0\\x=-5\text{, }&k=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{9k^{2}-14k+1}-3k-1}{2k}\text{; }x=-\frac{\sqrt{9k^{2}-14k+1}+3k+1}{2k}\text{, }&\left(k\neq 0\text{ and }k\leq \frac{7-2\sqrt{10}}{9}\right)\text{ or }k\geq \frac{2\sqrt{10}+7}{9}\\x=-5\text{, }&k=0\end{matrix}\right.$$