Solve frac{2+sqrt{4-x^2}}{x}=2

$$\frac{ 2+ \sqrt{ 4- { x }^{ 2 } } }{ x } =2$$

Solve for x

$$x = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

Subtract 2x from both sides.

$$2+\sqrt{4-x^{2}}-2x=0$$

Subtract 2-2x from both sides of the equation.

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

To find the opposite of 2-2x, find the opposite of each term.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-2+2x\right)^{2}.

$$4-x^{2}=4-8x+4x^{2}$$

Subtract 4 from both sides.

$$4-x^{2}-4=-8x+4x^{2}$$

Subtract 4 from 4 to get 0.

$$-x^{2}=-8x+4x^{2}$$

Add 8x to both sides.

$$-x^{2}+8x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$-x^{2}+8x-4x^{2}=0$$

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

$$-5x^{2}+8x=0$$

Factor out x.

$$x\left(-5x+8\right)=0$$

To find equation solutions, solve x=0 and -5x+8=0.

$$x=0 x=\frac{8}{5}$$

Substitute 0 for x in the equation \frac{2+\sqrt{4-x^{2}}}{x}=2. The expression is undefined.

$$\frac{2+\sqrt{4-0^{2}}}{0}=2$$

Substitute \frac{8}{5} for x in the equation \frac{2+\sqrt{4-x^{2}}}{x}=2.

$$\frac{2+\sqrt{4-\left(\frac{8}{5}\right)^{2}}}{\frac{8}{5}}=2$$

Simplify. The value x=\frac{8}{5} satisfies the equation.

$$2=2$$

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

$$x=\frac{8}{5}$$