Solve sqrt{c+1/2}+sqrt{b+1/2}=x

$$\sqrt { c + \frac { 1 } { 2 } } + \sqrt { b + \frac { 1 } { 2 } } = x$$

Solve for b

$$b=-\sqrt{4c+2}x+x^{2}+c$$

$$c\geq -\frac{1}{2}\text{ and }x-\frac{\sqrt{4c+2}}{2}\geq 0$$

Steps for Solving Radical Equations

Subtract \sqrt{c+\frac{1}{2}} from both sides of the equation.

$$\sqrt{b+\frac{1}{2}}+\sqrt{c+\frac{1}{2}}-\sqrt{c+\frac{1}{2}}=x-\sqrt{c+\frac{1}{2}}$$

Subtracting \sqrt{c+\frac{1}{2}} from itself leaves 0.

$$\sqrt{b+\frac{1}{2}}=x-\sqrt{c+\frac{1}{2}}$$

Subtract \sqrt{c+\frac{1}{2}} from x.

$$\sqrt{b+\frac{1}{2}}=x-\frac{\sqrt{4c+2}}{2}$$

Square both sides of the equation.

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

Subtract \frac{1}{2} from both sides of the equation.

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

Subtracting \frac{1}{2} from itself leaves 0.

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

Subtract \frac{1}{2} from \frac{\left(\sqrt{2}x-\sqrt{2c+1}\right)^{2}}{2}.

$$b=-\sqrt{4c+2}x+x^{2}+c$$

Solve for c

$$c=-\sqrt{4b+2}x+x^{2}+b$$

$$b\geq -\frac{1}{2}\text{ and }x-\frac{\sqrt{4b+2}}{2}\geq 0$$

Steps for Solving Radical Equations

Subtract \sqrt{b+\frac{1}{2}} from both sides of the equation.

$$\sqrt{c+\frac{1}{2}}+\sqrt{b+\frac{1}{2}}-\sqrt{b+\frac{1}{2}}=x-\sqrt{b+\frac{1}{2}}$$

Subtracting \sqrt{b+\frac{1}{2}} from itself leaves 0.

$$\sqrt{c+\frac{1}{2}}=x-\sqrt{b+\frac{1}{2}}$$

Subtract \sqrt{b+\frac{1}{2}} from x.

$$\sqrt{c+\frac{1}{2}}=x-\frac{\sqrt{4b+2}}{2}$$

Square both sides of the equation.

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

Subtract \frac{1}{2} from both sides of the equation.

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

Subtracting \frac{1}{2} from itself leaves 0.

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

Subtract \frac{1}{2} from \frac{\left(\sqrt{2}x-\sqrt{2b+1}\right)^{2}}{2}.

$$c=-\sqrt{4b+2}x+x^{2}+b$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\sqrt{4c+2}x+x^{2}+c\text{, }&arg(x-\frac{\sqrt{4c+2}}{2})<\pi \\b=-\frac{1}{2}=-0.5\text{, }&c=x^{2}-\frac{1}{2}\text{ and }|arg(-\sqrt{x^{2}})-arg(-x)|<\pi \end{matrix}\right.$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=-\sqrt{4b+2}x+x^{2}+b\text{, }&arg(x-\frac{\sqrt{4b+2}}{2})<\pi \\c=-\frac{1}{2}\text{, }&b=x^{2}-\frac{1}{2}\text{ and }|arg(-\sqrt{x^{2}})-arg(-x)|<\pi \end{matrix}\right.$$