Solve sqrt{x^2-1}-2x=0

$$\sqrt { x ^ { 2 } - 1 } - 2 x = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{3}i}{3}\approx 0.577350269i$$

Solution Steps

Subtract -2x from both sides of the equation.

$$\sqrt{x^{2}-1}=2x$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}-1}\right)^{2}=\left(2x\right)^{2}$$

Calculate \sqrt{x^{2}-1} to the power of 2 and get x^{2}-1.

$$x^{2}-1=\left(2x\right)^{2}$$

Expand \left(2x\right)^{2}.

$$x^{2}-1=2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-1=4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}-1-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$-3x^{2}=1$$

Divide both sides by -3.

$$x^{2}=-\frac{1}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{3}i}{3} x=-\frac{\sqrt{3}i}{3}$$

Substitute \frac{\sqrt{3}i}{3} for x in the equation \sqrt{x^{2}-1}-2x=0.

$$\sqrt{\left(\frac{\sqrt{3}i}{3}\right)^{2}-1}-2\times \frac{\sqrt{3}i}{3}=0$$

Simplify. The value x=\frac{\sqrt{3}i}{3} satisfies the equation.

$$0=0$$

Substitute -\frac{\sqrt{3}i}{3} for x in the equation \sqrt{x^{2}-1}-2x=0.

$$\sqrt{\left(-\frac{\sqrt{3}i}{3}\right)^{2}-1}-2\left(-\frac{\sqrt{3}i}{3}\right)=0$$

Simplify. The value x=-\frac{\sqrt{3}i}{3} does not satisfy the equation.

$$\frac{4}{3}i\times 3^{\frac{1}{2}}=0$$

Equation \sqrt{x^{2}-1}=2x has a unique solution.

$$x=\frac{\sqrt{3}i}{3}$$