Solve x-2/4=x+2/2x

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

Solve for x

$$x=2\sqrt{2}+2\approx 4.828427125$$

$$x=2-2\sqrt{2}\approx -0.828427125$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x, the least common multiple of 4,2x.

$$x\left(x-2\right)=2\left(x+2\right)$$

Use the distributive property to multiply x by x-2.

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

Use the distributive property to multiply 2 by x+2.

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

Subtract 2x from both sides.

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

Combine -2x and -2x to get -4x.

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

Subtract 4 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$x=\frac{-\left(-4\right)±\sqrt{16+16}}{2}$$

Add 16 to 16.

$$x=\frac{-\left(-4\right)±\sqrt{32}}{2}$$

Take the square root of 32.

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

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{2}}{2}$$

Now solve the equation x=\frac{4±4\sqrt{2}}{2} when ± is plus. Add 4 to 4\sqrt{2}.

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

Divide 4+4\sqrt{2} by 2.

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

Now solve the equation x=\frac{4±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from 4.

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

Divide 4-4\sqrt{2} by 2.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x, the least common multiple of 4,2x.

$$x\left(x-2\right)=2\left(x+2\right)$$

Use the distributive property to multiply x by x-2.

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

Use the distributive property to multiply 2 by x+2.

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

Subtract 2x from both sides.

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

Combine -2x and -2x to get -4x.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add 4 to 4.

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

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{8}$$

Simplify.

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

Add 2 to both sides of the equation.

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