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

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

Solve for x

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

Solution Steps

Subtract 2x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Calculate -1 to the power of 2 and get 1.

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

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

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

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

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

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

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

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}.

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

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -3.

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

Fraction \frac{-1}{-3} can be simplified to \frac{1}{3} by removing the negative sign from both the numerator and the denominator.

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

Take the square root of both sides of the equation.

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

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

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

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

$$0=0$$

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

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

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

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

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

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