Solve sqrt{x^2-a^2}=x+a

$$\sqrt { x ^ { 2 } - a ^ { 2 } } = x + a$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-a\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=0\text{ and }arg(x)<\pi \end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x^{2}-a^{2}} to the power of 2 and get x^{2}-a^{2}.

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

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

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

Subtract x^{2} from both sides.

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

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

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

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

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

Subtract a^{2} from both sides.

$$2xa=-a^{2}-a^{2}$$

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

$$2xa=-2a^{2}$$

The equation is in standard form.

$$2ax=-2a^{2}$$

Divide both sides by 2a.

$$\frac{2ax}{2a}=-\frac{2a^{2}}{2a}$$

Dividing by 2a undoes the multiplication by 2a.

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

Divide -2a^{2} by 2a.

$$x=-a$$

Substitute -a for x in the equation \sqrt{x^{2}-a^{2}}=x+a.

$$\sqrt{\left(-a\right)^{2}-a^{2}}=-a+a$$

Simplify. The value x=-a satisfies the equation.

$$0=0$$

Equation \sqrt{x^{2}-a^{2}}=x+a has a unique solution.

$$x=-a$$

Solve for x

$$\left\{\begin{matrix}x=-a\text{, }&a\neq 0\\x\geq 0\text{, }&a=0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x^{2}-a^{2}} to the power of 2 and get x^{2}-a^{2}.

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

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

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

Subtract x^{2} from both sides.

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

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

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

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

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

Subtract a^{2} from both sides.

$$2xa=-a^{2}-a^{2}$$

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

$$2xa=-2a^{2}$$

The equation is in standard form.

$$2ax=-2a^{2}$$

Divide both sides by 2a.

$$\frac{2ax}{2a}=-\frac{2a^{2}}{2a}$$

Dividing by 2a undoes the multiplication by 2a.

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

Divide -2a^{2} by 2a.

$$x=-a$$

Substitute -a for x in the equation \sqrt{x^{2}-a^{2}}=x+a.

$$\sqrt{\left(-a\right)^{2}-a^{2}}=-a+a$$

Simplify. The value x=-a satisfies the equation.

$$0=0$$

Equation \sqrt{x^{2}-a^{2}}=x+a has a unique solution.

$$x=-a$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=-x\text{, }&\text{unconditionally}\\a=0\text{, }&arg(x)<\pi \end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}\\a=-x\text{, }&\text{unconditionally}\\a=0\text{, }&x\geq 0\end{matrix}\right.$$