Solve x/5=x+2/x-2-4/5

$$\frac { x } { 5 } = \frac { x + 2 } { x - 2 } - \frac { 4 } { 5 }$$

Solve for x

$$x=-3$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-2\right), the least common multiple of 5,x-2.

$$\left(x-2\right)x=5\left(x+2\right)+5\left(x-2\right)\left(-\frac{4}{5}\right)$$

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

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

Use the distributive property to multiply 5 by x+2.

$$x^{2}-2x=5x+10+5\left(x-2\right)\left(-\frac{4}{5}\right)$$

Multiply 5 and -\frac{4}{5} to get -4.

$$x^{2}-2x=5x+10-4\left(x-2\right)$$

Use the distributive property to multiply -4 by x-2.

$$x^{2}-2x=5x+10-4x+8$$

Combine 5x and -4x to get x.

$$x^{2}-2x=x+10+8$$

Add 10 and 8 to get 18.

$$x^{2}-2x=x+18$$

Subtract x from both sides.

$$x^{2}-2x-x=18$$

Combine -2x and -x to get -3x.

$$x^{2}-3x=18$$

Subtract 18 from both sides.

$$x^{2}-3x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-18\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2}$$

Add 9 to 72.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$x=\frac{-\left(-3\right)±9}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±9}{2}$$

Now solve the equation x=\frac{3±9}{2} when ± is plus. Add 3 to 9.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{3±9}{2} when ± is minus. Subtract 9 from 3.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=6 x=-3$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-2\right), the least common multiple of 5,x-2.

$$\left(x-2\right)x=5\left(x+2\right)+5\left(x-2\right)\left(-\frac{4}{5}\right)$$

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

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

Use the distributive property to multiply 5 by x+2.

$$x^{2}-2x=5x+10+5\left(x-2\right)\left(-\frac{4}{5}\right)$$

Multiply 5 and -\frac{4}{5} to get -4.

$$x^{2}-2x=5x+10-4\left(x-2\right)$$

Use the distributive property to multiply -4 by x-2.

$$x^{2}-2x=5x+10-4x+8$$

Combine 5x and -4x to get x.

$$x^{2}-2x=x+10+8$$

Add 10 and 8 to get 18.

$$x^{2}-2x=x+18$$

Subtract x from both sides.

$$x^{2}-2x-x=18$$

Combine -2x and -x to get -3x.

$$x^{2}-3x=18$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=18+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=18+\frac{9}{4}$$

Add 18 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{81}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

$$x=6 x=-3$$