Solve (3a+1)x^2+(a+2)x+1=0

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

Solve for a

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

$$x\neq -\frac{1}{3}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

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

$$3ax^{2}+x^{2}+\left(a+2\right)x+1=0$$

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

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

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

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

Subtract 2x from both sides.

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

Subtract 1 from both sides.

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

Combine all terms containing a.

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

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

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

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

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

Divide -\left(x+1\right)^{2} by 3x^{2}+x.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{a\left(a-8\right)}-a-2}{2\left(3a+1\right)}\text{; }x=-\frac{\sqrt{a\left(a-8\right)}+a+2}{2\left(3a+1\right)}\text{, }&a\neq -\frac{1}{3}\\x=-\frac{3}{5}\text{, }&a=-\frac{1}{3}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{a\left(a-8\right)}-a-2}{2\left(3a+1\right)}\text{; }x=-\frac{\sqrt{a\left(a-8\right)}+a+2}{2\left(3a+1\right)}\text{, }&\left(a\neq -\frac{1}{3}\text{ and }a\leq 0\right)\text{ or }a\geq 8\\x=-\frac{3}{5}\text{, }&a=-\frac{1}{3}\end{matrix}\right.$$