Solve x=sqrt{4x^2-1}

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

Solve for x

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

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{4x^{2}-1} to the power of 2 and get 4x^{2}-1.

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

Subtract 4x^{2} from both sides.

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

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

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

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 x=\sqrt{4x^{2}-1}.

$$\frac{\sqrt{3}}{3}=\sqrt{4\times \left(\frac{\sqrt{3}}{3}\right)^{2}-1}$$

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

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

Substitute -\frac{\sqrt{3}}{3} for x in the equation x=\sqrt{4x^{2}-1}.

$$-\frac{\sqrt{3}}{3}=\sqrt{4\left(-\frac{\sqrt{3}}{3}\right)^{2}-1}$$

Simplify. The value x=-\frac{\sqrt{3}}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

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

Equation x=\sqrt{4x^{2}-1} has a unique solution.

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