Solve sqrt{x^2+3}=2x?

$$\sqrt { x ^ { 2 } + 3 } = 2 x ?$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}+3}\right)^{2}=\left(2x\right)^{2}$$

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

$$x^{2}+3=\left(2x\right)^{2}$$

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

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4x^{2} from both sides.

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

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

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

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -3.

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

Divide -3 by -3 to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

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

$$\sqrt{1^{2}+3}=2\times 1$$

Simplify. The value x=1 satisfies the equation.

$$2=2$$

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

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

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{x^{2}+3}=2x has a unique solution.

$$x=1$$