Solve 1/x=2ax+1

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

Solve for a

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

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$1=2axx+x$$

Multiply x and x to get x^{2}.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract x from both sides.

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

The equation is in standard form.

$$2x^{2}a=1-x$$

Divide both sides by 2x^{2}.

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{8a+1}+1}{4a}\text{; }x=-\frac{-\sqrt{8a+1}+1}{4a}\text{, }&a\neq 0\\x=1\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{8a+1}+1}{4a}\text{; }x=-\frac{-\sqrt{8a+1}+1}{4a}\text{, }&a\neq 0\text{ and }a\geq -\frac{1}{8}\\x=1\text{, }&a=0\end{matrix}\right.$$