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

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

Solve for a

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

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

Steps for Solving Linear Equation

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

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

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

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

Subtract 1 from both sides.

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

Combine all terms containing a.

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

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

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

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

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

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

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

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{a^{2}+4}-a-2}{2a}\text{; }x=-\frac{\sqrt{a^{2}+4}+a+2}{2a}\text{, }&a\neq 0\\x=-\frac{1}{2}\text{, }&a=0\end{matrix}\right.$$