Solve (3k+1)x^2+2(k+1)x+k+8=0

$$( 3 k + 1 ) x ^ { 2 } + 2 ( k + 1 ) x + k + 8 = 0$$

Solve for k (complex solution)

$$k=-\frac{x^{2}+2x+8}{3x^{2}+2x+1}$$

$$x\neq \frac{-1+\sqrt{2}i}{3}\text{ and }x\neq \frac{-\sqrt{2}i-1}{3}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3k+1 by x^{2}.

$$3kx^{2}+x^{2}+2\left(k+1\right)x+k+8=0$$

Use the distributive property to multiply 2 by k+1.

$$3kx^{2}+x^{2}+\left(2k+2\right)x+k+8=0$$

Use the distributive property to multiply 2k+2 by x.

$$3kx^{2}+x^{2}+2kx+2x+k+8=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$3kx^{2}+2kx+2x+k+8=-x^{2}$$

Subtract 2x from both sides.

$$3kx^{2}+2kx+k+8=-x^{2}-2x$$

Subtract 8 from both sides.

$$3kx^{2}+2kx+k=-x^{2}-2x-8$$

Combine all terms containing k.

$$\left(3x^{2}+2x+1\right)k=-x^{2}-2x-8$$

Divide both sides by 3x^{2}+2x+1.

$$\frac{\left(3x^{2}+2x+1\right)k}{3x^{2}+2x+1}=\frac{-x^{2}-2x-8}{3x^{2}+2x+1}$$

Dividing by 3x^{2}+2x+1 undoes the multiplication by 3x^{2}+2x+1.

$$k=\frac{-x^{2}-2x-8}{3x^{2}+2x+1}$$

Divide -x^{2}-2x-8 by 3x^{2}+2x+1.

$$k=-\frac{x^{2}+2x+8}{3x^{2}+2x+1}$$

Solve for k

$$k=-\frac{x^{2}+2x+8}{3x^{2}+2x+1}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3k+1 by x^{2}.

$$3kx^{2}+x^{2}+2\left(k+1\right)x+k+8=0$$

Use the distributive property to multiply 2 by k+1.

$$3kx^{2}+x^{2}+\left(2k+2\right)x+k+8=0$$

Use the distributive property to multiply 2k+2 by x.

$$3kx^{2}+x^{2}+2kx+2x+k+8=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$3kx^{2}+2kx+2x+k+8=-x^{2}$$

Subtract 2x from both sides.

$$3kx^{2}+2kx+k+8=-x^{2}-2x$$

Subtract 8 from both sides.

$$3kx^{2}+2kx+k=-x^{2}-2x-8$$

Combine all terms containing k.

$$\left(3x^{2}+2x+1\right)k=-x^{2}-2x-8$$

Divide both sides by 3x^{2}+2x+1.

$$\frac{\left(3x^{2}+2x+1\right)k}{3x^{2}+2x+1}=\frac{-x^{2}-2x-8}{3x^{2}+2x+1}$$

Dividing by 3x^{2}+2x+1 undoes the multiplication by 3x^{2}+2x+1.

$$k=\frac{-x^{2}-2x-8}{3x^{2}+2x+1}$$

Divide -x^{2}-2x-8 by 3x^{2}+2x+1.

$$k=-\frac{x^{2}+2x+8}{3x^{2}+2x+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{-2k^{2}-23k-7}-k-1}{3k+1}\text{; }x=-\frac{\sqrt{-2k^{2}-23k-7}+k+1}{3k+1}\text{, }&k\neq -\frac{1}{3}\\x=-\frac{23}{4}\text{, }&k=-\frac{1}{3}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{-2k^{2}-23k-7}-k-1}{3k+1}\text{; }x=-\frac{\sqrt{-2k^{2}-23k-7}+k+1}{3k+1}\text{, }&k\neq -\frac{1}{3}\text{ and }k\geq \frac{-\sqrt{473}-23}{4}\text{ and }k\leq \frac{\sqrt{473}-23}{4}\\x=-\frac{23}{4}\text{, }&k=-\frac{1}{3}\end{matrix}\right.$$