Solve (a+c)x+x^2=(x+a)^2

$$( a + c ) x + x ^ { 2 } = ( x + a ) ^ { 2 }$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=\frac{a\left(x+a\right)}{x}\text{, }&x\neq 0\\c\in \mathrm{C}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+cx+x^{2}=\left(x+a\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+a\right)^{2}.

$$ax+cx+x^{2}=x^{2}+2xa+a^{2}$$

Subtract ax from both sides.

$$cx+x^{2}=x^{2}+2xa+a^{2}-ax$$

Combine 2xa and -ax to get xa.

$$cx+x^{2}=x^{2}+xa+a^{2}$$

Subtract x^{2} from both sides.

$$cx=x^{2}+xa+a^{2}-x^{2}$$

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

$$cx=xa+a^{2}$$

The equation is in standard form.

$$xc=ax+a^{2}$$

Divide both sides by x.

$$\frac{xc}{x}=\frac{a\left(x+a\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$c=\frac{a\left(x+a\right)}{x}$$

Solve for c

$$\left\{\begin{matrix}c=\frac{a\left(x+a\right)}{x}\text{, }&x\neq 0\\c\in \mathrm{R}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+cx+x^{2}=\left(x+a\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+a\right)^{2}.

$$ax+cx+x^{2}=x^{2}+2xa+a^{2}$$

Subtract ax from both sides.

$$cx+x^{2}=x^{2}+2xa+a^{2}-ax$$

Combine 2xa and -ax to get xa.

$$cx+x^{2}=x^{2}+xa+a^{2}$$

Subtract x^{2} from both sides.

$$cx=x^{2}+xa+a^{2}-x^{2}$$

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

$$cx=xa+a^{2}$$

The equation is in standard form.

$$xc=ax+a^{2}$$

Divide both sides by x.

$$\frac{xc}{x}=\frac{a\left(x+a\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$c=\frac{a\left(x+a\right)}{x}$$

Solve for a (complex solution)

$$a=\frac{-\sqrt{x\left(x+4c\right)}-x}{2}$$

$$a=\frac{\sqrt{x\left(x+4c\right)}-x}{2}$$

Solve for a

$$a=\frac{-\sqrt{x\left(x+4c\right)}-x}{2}$$

$$a=\frac{\sqrt{x\left(x+4c\right)}-x}{2}\text{, }\left(x\leq 0\text{ or }c\geq -\frac{x}{4}\right)\text{ and }\left(x\geq 0\text{ or }c\leq -\frac{x}{4}\right)$$