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

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

Solve for x

$$x=\sqrt{3}\approx 1.732050808$$

Solution Steps

Square both sides of the equation.

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

Calculate 3 to the power of 2 and get 9.

$$\left(\sqrt{9+x^{2}}\right)^{2}=\left(2x\right)^{2}$$

Calculate \sqrt{9+x^{2}} to the power of 2 and get 9+x^{2}.

$$9+x^{2}=\left(2x\right)^{2}$$

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

$$9+x^{2}=2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$9+x^{2}=4x^{2}$$

Subtract 4x^{2} from both sides.

$$9+x^{2}-4x^{2}=0$$

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

$$9-3x^{2}=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$-3x^{2}=-9$$

Divide both sides by -3.

$$x^{2}=\frac{-9}{-3}$$

Divide -9 by -3 to get 3.

$$x^{2}=3$$

Take the square root of both sides of the equation.

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

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

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

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

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

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

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

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

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

Equation \sqrt{x^{2}+9}=2x has a unique solution.

$$x=\sqrt{3}$$