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

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{x\left(x+2b\right)}{2\left(x+b\right)}\text{, }&b\neq -x\\a\in \mathrm{C}\text{, }&x=0\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b+x.

$$a^{2}+b^{2}=x^{2}+2ax+2bx+2ab+a^{2}+b^{2}$$

Subtract 2ax from both sides.

$$a^{2}+b^{2}-2ax=x^{2}+2bx+2ab+a^{2}+b^{2}$$

Subtract 2ab from both sides.

$$a^{2}+b^{2}-2ax-2ab=x^{2}+2bx+a^{2}+b^{2}$$

Subtract a^{2} from both sides.

$$a^{2}+b^{2}-2ax-2ab-a^{2}=x^{2}+2bx+b^{2}$$

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

$$b^{2}-2ax-2ab=x^{2}+2bx+b^{2}$$

Subtract b^{2} from both sides.

$$-2ax-2ab=x^{2}+2bx+b^{2}-b^{2}$$

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

$$-2ax-2ab=x^{2}+2bx$$

Combine all terms containing a.

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

Divide both sides by -2b-2x.

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

Dividing by -2b-2x undoes the multiplication by -2b-2x.

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

Divide x\left(x+2b\right) by -2b-2x.

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

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{x\left(x+2a\right)}{2\left(x+a\right)}\text{, }&a\neq -x\\b\in \mathrm{C}\text{, }&x=0\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b+x.

$$a^{2}+b^{2}=x^{2}+2ax+2bx+2ab+a^{2}+b^{2}$$

Subtract 2bx from both sides.

$$a^{2}+b^{2}-2bx=x^{2}+2ax+2ab+a^{2}+b^{2}$$

Subtract 2ab from both sides.

$$a^{2}+b^{2}-2bx-2ab=x^{2}+2ax+a^{2}+b^{2}$$

Subtract b^{2} from both sides.

$$a^{2}+b^{2}-2bx-2ab-b^{2}=x^{2}+2ax+a^{2}$$

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

$$a^{2}-2bx-2ab=x^{2}+2ax+a^{2}$$

Subtract a^{2} from both sides.

$$-2bx-2ab=x^{2}+2ax+a^{2}-a^{2}$$

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

$$-2bx-2ab=x^{2}+2ax$$

Combine all terms containing b.

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

Divide both sides by -2a-2x.

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

Dividing by -2a-2x undoes the multiplication by -2a-2x.

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

Divide x\left(x+2a\right) by -2a-2x.

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{x\left(x+2b\right)}{2\left(x+b\right)}\text{, }&b\neq -x\\a\in \mathrm{R}\text{, }&x=0\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b+x.

$$a^{2}+b^{2}=x^{2}+2ax+2bx+2ab+a^{2}+b^{2}$$

Subtract 2ax from both sides.

$$a^{2}+b^{2}-2ax=x^{2}+2bx+2ab+a^{2}+b^{2}$$

Subtract 2ab from both sides.

$$a^{2}+b^{2}-2ax-2ab=x^{2}+2bx+a^{2}+b^{2}$$

Subtract a^{2} from both sides.

$$a^{2}+b^{2}-2ax-2ab-a^{2}=x^{2}+2bx+b^{2}$$

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

$$b^{2}-2ax-2ab=x^{2}+2bx+b^{2}$$

Subtract b^{2} from both sides.

$$-2ax-2ab=x^{2}+2bx+b^{2}-b^{2}$$

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

$$-2ax-2ab=x^{2}+2bx$$

Combine all terms containing a.

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

Divide both sides by -2b-2x.

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

Dividing by -2b-2x undoes the multiplication by -2b-2x.

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

Divide x\left(x+2b\right) by -2b-2x.

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

Solve for b

$$\left\{\begin{matrix}b=-\frac{x\left(x+2a\right)}{2\left(x+a\right)}\text{, }&a\neq -x\\b\in \mathrm{R}\text{, }&x=0\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b+x.

$$a^{2}+b^{2}=x^{2}+2ax+2bx+2ab+a^{2}+b^{2}$$

Subtract 2bx from both sides.

$$a^{2}+b^{2}-2bx=x^{2}+2ax+2ab+a^{2}+b^{2}$$

Subtract 2ab from both sides.

$$a^{2}+b^{2}-2bx-2ab=x^{2}+2ax+a^{2}+b^{2}$$

Subtract b^{2} from both sides.

$$a^{2}+b^{2}-2bx-2ab-b^{2}=x^{2}+2ax+a^{2}$$

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

$$a^{2}-2bx-2ab=x^{2}+2ax+a^{2}$$

Subtract a^{2} from both sides.

$$-2bx-2ab=x^{2}+2ax+a^{2}-a^{2}$$

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

$$-2bx-2ab=x^{2}+2ax$$

Combine all terms containing b.

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

Divide both sides by -2a-2x.

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

Dividing by -2a-2x undoes the multiplication by -2a-2x.

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

Divide x\left(x+2a\right) by -2a-2x.

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