Solve x^2+3ax+3a+1=0

$$x ^ { 2 } + 3 a x + 3 a + 1 = 0$$

Solve for a

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

$$x\neq -1$$

Steps for Solving Linear Equation

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

$$3ax+3a+1=-x^{2}$$

Subtract 1 from both sides.

$$3ax+3a=-x^{2}-1$$

Combine all terms containing a.

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

Divide both sides by 3x+3.

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

Dividing by 3x+3 undoes the multiplication by 3x+3.

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

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

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

Solve for x (complex solution)

$$x=\frac{\sqrt{9a^{2}-12a-4}-3a}{2}$$

$$x=\frac{-\sqrt{9a^{2}-12a-4}-3a}{2}$$

Solve for x

$$x=\frac{\sqrt{9a^{2}-12a-4}-3a}{2}$$

$$x=\frac{-\sqrt{9a^{2}-12a-4}-3a}{2}\text{, }a\geq \frac{2\sqrt{2}+2}{3}\text{ or }a\leq \frac{2-2\sqrt{2}}{3}$$