Solve sqrt{x+6}-sqrt{x-6}=2sqrt{3}

$$\sqrt { x + 6 } - \sqrt { x - 6 } = 2 \sqrt { 3 }$$

Solve for x

$$x=6$$

Solution Steps

Subtract -\sqrt{x-6} from both sides of the equation.

$$\sqrt{x+6}=2\sqrt{3}+\sqrt{x-6}$$

Square both sides of the equation.

$$\left(\sqrt{x+6}\right)^{2}=\left(2\sqrt{3}+\sqrt{x-6}\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$x+6=\left(2\sqrt{3}+\sqrt{x-6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{3}+\sqrt{x-6}\right)^{2}.

$$x+6=4\left(\sqrt{3}\right)^{2}+4\sqrt{3}\sqrt{x-6}+\left(\sqrt{x-6}\right)^{2}$$

The square of \sqrt{3} is 3.

$$x+6=4\times 3+4\sqrt{3}\sqrt{x-6}+\left(\sqrt{x-6}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$x+6=12+4\sqrt{3}\sqrt{x-6}+\left(\sqrt{x-6}\right)^{2}$$

Calculate \sqrt{x-6} to the power of 2 and get x-6.

$$x+6=12+4\sqrt{3}\sqrt{x-6}+x-6$$

Subtract 6 from 12 to get 6.

$$x+6=6+4\sqrt{3}\sqrt{x-6}+x$$

Subtract 4\sqrt{3}\sqrt{x-6} from both sides.

$$x+6-4\sqrt{3}\sqrt{x-6}=6+x$$

Subtract x from both sides.

$$x+6-4\sqrt{3}\sqrt{x-6}-x=6$$

Combine x and -x to get 0.

$$6-4\sqrt{3}\sqrt{x-6}=6$$

Subtract 6 from both sides.

$$-4\sqrt{3}\sqrt{x-6}=6-6$$

Subtract 6 from 6 to get 0.

$$-4\sqrt{3}\sqrt{x-6}=0$$

Divide both sides by -4\sqrt{3}.

$$\frac{\left(-4\sqrt{3}\right)\sqrt{x-6}}{-4\sqrt{3}}=\frac{0}{-4\sqrt{3}}$$

Dividing by -4\sqrt{3} undoes the multiplication by -4\sqrt{3}.

$$\sqrt{x-6}=\frac{0}{-4\sqrt{3}}$$

Divide 0 by -4\sqrt{3}.

$$\sqrt{x-6}=0$$

Square both sides of the equation.

$$x-6=0$$

Add 6 to both sides of the equation.

$$x-6-\left(-6\right)=-\left(-6\right)$$

Subtracting -6 from itself leaves 0.

$$x=-\left(-6\right)$$

Subtract -6 from 0.

$$x=6$$

Substitute 6 for x in the equation \sqrt{x+6}-\sqrt{x-6}=2\sqrt{3}.

$$\sqrt{6+6}-\sqrt{6-6}=2\sqrt{3}$$

Simplify. The value x=6 satisfies the equation.

$$2\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}$$

Equation \sqrt{x+6}=\sqrt{x-6}+2\sqrt{3} has a unique solution.

$$x=6$$