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

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

Solve for x

$$x = \frac{2 \sqrt{3}}{3} \approx 1.154700538$$

Solution Steps

Subtract -x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(2\sqrt{x^{2}-1}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x^{2}-1}\right)^{2}=x^{2}$$

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

$$4\left(x^{2}-1\right)=x^{2}$$

Use the distributive property to multiply 4 by x^{2}-1.

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

Subtract x^{2} from both sides.

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

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

$$3x^{2}-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$3x^{2}=4$$

Divide both sides by 3.

$$x^{2}=\frac{4}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{2\sqrt{3}}{3} x=-\frac{2\sqrt{3}}{3}$$

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

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

Simplify. The value x=\frac{2\sqrt{3}}{3} satisfies the equation.

$$0=0$$

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

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

Simplify. The value x=-\frac{2\sqrt{3}}{3} does not satisfy the equation.

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

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

$$x=\frac{2\sqrt{3}}{3}$$