Solve sqrt{4x^2-1}-sqrt{5x^2-3}=0

$$\sqrt{ 4 { x }^{ 2 } -1 } - \sqrt{ 5 { x }^{ 2 } -3 } =0$$

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

$$x=-\sqrt{2}\approx -1.414213562$$

Solution Steps

Subtract -\sqrt{5x^{2}-3} from both sides of the equation.

$$\sqrt{4x^{2}-1}=\sqrt{5x^{2}-3}$$

Square both sides of the equation.

$$\left(\sqrt{4x^{2}-1}\right)^{2}=\left(\sqrt{5x^{2}-3}\right)^{2}$$

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

$$4x^{2}-1=\left(\sqrt{5x^{2}-3}\right)^{2}$$

Calculate \sqrt{5x^{2}-3} to the power of 2 and get 5x^{2}-3.

$$4x^{2}-1=5x^{2}-3$$

Subtract 5x^{2} from both sides.

$$4x^{2}-1-5x^{2}=-3$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

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

Add 1 to both sides.

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

Add -3 and 1 to get -2.

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

Divide both sides by -1.

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

Fraction \frac{-2}{-1} can be simplified to 2 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=2$$

Take the square root of both sides of the equation.

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

Substitute \sqrt{2} for x in the equation \sqrt{4x^{2}-1}-\sqrt{5x^{2}-3}=0.

$$\sqrt{4\left(\sqrt{2}\right)^{2}-1}-\sqrt{5\left(\sqrt{2}\right)^{2}-3}=0$$

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

$$0=0$$

Substitute -\sqrt{2} for x in the equation \sqrt{4x^{2}-1}-\sqrt{5x^{2}-3}=0.

$$\sqrt{4\left(-\sqrt{2}\right)^{2}-1}-\sqrt{5\left(-\sqrt{2}\right)^{2}-3}=0$$

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

$$0=0$$

List all solutions of \sqrt{4x^{2}-1}=\sqrt{5x^{2}-3}.

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