Solve (x+m/2L)^2=-n/L+m^2/4L^2

$$( x + \frac { m } { 2 L } ) ^ { 2 } = - \frac { n } { L } + \frac { m ^ { 2 } } { 4 L ^ { 2 } }$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-Lx-\frac{n}{x}\text{, }&x\neq 0\text{ and }L\neq 0\\m\in \mathrm{C}\text{, }&n=0\text{ and }x=0\text{ and }L\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4L^{2}, the least common multiple of L,4L^{2}.

$$4L^{2}\left(x+\frac{m}{2L}\right)^{2}=-4Ln+m^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2L}{2L}.

$$4L^{2}\left(\frac{x\times 2L}{2L}+\frac{m}{2L}\right)^{2}=-4Ln+m^{2}$$

Since \frac{x\times 2L}{2L} and \frac{m}{2L} have the same denominator, add them by adding their numerators.

$$4L^{2}\times \left(\frac{x\times 2L+m}{2L}\right)^{2}=-4Ln+m^{2}$$

To raise \frac{x\times 2L+m}{2L} to a power, raise both numerator and denominator to the power and then divide.

$$4L^{2}\times \frac{\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}}=-4Ln+m^{2}$$

Express 4\times \frac{\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}} as a single fraction.

$$\frac{4\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}}L^{2}=-4Ln+m^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x\times 2L+m\right)^{2}.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{\left(2L\right)^{2}}L^{2}=-4Ln+m^{2}$$

Expand \left(2L\right)^{2}.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{2^{2}L^{2}}L^{2}=-4Ln+m^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{4L^{2}}L^{2}=-4Ln+m^{2}$$

Cancel out 4 in both numerator and denominator.

$$\frac{4L^{2}x^{2}+4Lmx+m^{2}}{L^{2}}L^{2}=-4Ln+m^{2}$$

Express \frac{4L^{2}x^{2}+4Lmx+m^{2}}{L^{2}}L^{2} as a single fraction.

$$\frac{\left(4L^{2}x^{2}+4Lmx+m^{2}\right)L^{2}}{L^{2}}=-4Ln+m^{2}$$

Cancel out L^{2} in both numerator and denominator.

$$4L^{2}x^{2}+4Lmx+m^{2}=-4Ln+m^{2}$$

Subtract m^{2} from both sides.

$$4L^{2}x^{2}+4Lmx+m^{2}-m^{2}=-4Ln$$

Combine m^{2} and -m^{2} to get 0.

$$4L^{2}x^{2}+4Lmx=-4Ln$$

Subtract 4L^{2}x^{2} from both sides.

$$4Lmx=-4Ln-4L^{2}x^{2}$$

The equation is in standard form.

$$4Lxm=-4L^{2}x^{2}-4Ln$$

Divide both sides by 4Lx.

$$\frac{4Lxm}{4Lx}=-\frac{4L\left(Lx^{2}+n\right)}{4Lx}$$

Dividing by 4Lx undoes the multiplication by 4Lx.

$$m=-\frac{4L\left(Lx^{2}+n\right)}{4Lx}$$

Divide -4L\left(n+Lx^{2}\right) by 4Lx.

$$m=-Lx-\frac{n}{x}$$

Solve for m

$$\left\{\begin{matrix}m=-Lx-\frac{n}{x}\text{, }&x\neq 0\text{ and }L\neq 0\\m\in \mathrm{R}\text{, }&n=0\text{ and }x=0\text{ and }L\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4L^{2}, the least common multiple of L,4L^{2}.

$$4L^{2}\left(x+\frac{m}{2L}\right)^{2}=-4Ln+m^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2L}{2L}.

$$4L^{2}\left(\frac{x\times 2L}{2L}+\frac{m}{2L}\right)^{2}=-4Ln+m^{2}$$

Since \frac{x\times 2L}{2L} and \frac{m}{2L} have the same denominator, add them by adding their numerators.

$$4L^{2}\times \left(\frac{x\times 2L+m}{2L}\right)^{2}=-4Ln+m^{2}$$

To raise \frac{x\times 2L+m}{2L} to a power, raise both numerator and denominator to the power and then divide.

$$4L^{2}\times \frac{\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}}=-4Ln+m^{2}$$

Express 4\times \frac{\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}} as a single fraction.

$$\frac{4\left(x\times 2L+m\right)^{2}}{\left(2L\right)^{2}}L^{2}=-4Ln+m^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x\times 2L+m\right)^{2}.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{\left(2L\right)^{2}}L^{2}=-4Ln+m^{2}$$

Expand \left(2L\right)^{2}.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{2^{2}L^{2}}L^{2}=-4Ln+m^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\left(4x^{2}L^{2}+4xLm+m^{2}\right)}{4L^{2}}L^{2}=-4Ln+m^{2}$$

Cancel out 4 in both numerator and denominator.

$$\frac{4L^{2}x^{2}+4Lmx+m^{2}}{L^{2}}L^{2}=-4Ln+m^{2}$$

Express \frac{4L^{2}x^{2}+4Lmx+m^{2}}{L^{2}}L^{2} as a single fraction.

$$\frac{\left(4L^{2}x^{2}+4Lmx+m^{2}\right)L^{2}}{L^{2}}=-4Ln+m^{2}$$

Cancel out L^{2} in both numerator and denominator.

$$4L^{2}x^{2}+4Lmx+m^{2}=-4Ln+m^{2}$$

Subtract m^{2} from both sides.

$$4L^{2}x^{2}+4Lmx+m^{2}-m^{2}=-4Ln$$

Combine m^{2} and -m^{2} to get 0.

$$4L^{2}x^{2}+4Lmx=-4Ln$$

Subtract 4L^{2}x^{2} from both sides.

$$4Lmx=-4Ln-4L^{2}x^{2}$$

The equation is in standard form.

$$4Lxm=-4L^{2}x^{2}-4Ln$$

Divide both sides by 4Lx.

$$\frac{4Lxm}{4Lx}=-\frac{4L\left(Lx^{2}+n\right)}{4Lx}$$

Dividing by 4Lx undoes the multiplication by 4Lx.

$$m=-\frac{4L\left(Lx^{2}+n\right)}{4Lx}$$

Divide -4L\left(n+Lx^{2}\right) by 4Lx.

$$m=-Lx-\frac{n}{x}$$

Solve for L

$$\left\{\begin{matrix}L=-\frac{mx+n}{x^{2}}\text{, }&\left(n\neq 0\text{ or }m\neq 0\right)\text{ and }\left(m=0\text{ or }x\neq -\frac{n}{m}\right)\text{ and }x\neq 0\text{ and }n\neq -mx\\L\neq 0\text{, }&n=0\text{ and }x=0\end{matrix}\right.$$