Solve x-2/8-x-2/x=1/4

$$\frac { x - 2 } { 8 } - \frac { x - 2 } { x } = \frac { 1 } { 4 }$$

Solve for x

$$x=2\sqrt{5}+6\approx 10.472135955$$

$$x=6-2\sqrt{5}\approx 1.527864045$$

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 8x, the least common multiple of 8,x,4.

$$x\left(x-2\right)-8\left(x-2\right)=2x$$

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

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

Use the distributive property to multiply -8 by x-2.

$$x^{2}-2x-8x+16=2x$$

Combine -2x and -8x to get -10x.

$$x^{2}-10x+16=2x$$

Subtract 2x from both sides.

$$x^{2}-10x+16-2x=0$$

Combine -10x and -2x to get -12x.

$$x^{2}-12x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 16}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 16}}{2}$$

Multiply -4 times 16.

$$x=\frac{-\left(-12\right)±\sqrt{144-64}}{2}$$

Add 144 to -64.

$$x=\frac{-\left(-12\right)±\sqrt{80}}{2}$$

Take the square root of 80.

$$x=\frac{-\left(-12\right)±4\sqrt{5}}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{5}}{2}$$

Now solve the equation x=\frac{12±4\sqrt{5}}{2} when ± is plus. Add 12 to 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+12}{2}$$

Divide 12+4\sqrt{5} by 2.

$$x=2\sqrt{5}+6$$

Now solve the equation x=\frac{12±4\sqrt{5}}{2} when ± is minus. Subtract 4\sqrt{5} from 12.

$$x=\frac{12-4\sqrt{5}}{2}$$

Divide 12-4\sqrt{5} by 2.

$$x=6-2\sqrt{5}$$

The equation is now solved.

$$x=2\sqrt{5}+6 x=6-2\sqrt{5}$$

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 8x, the least common multiple of 8,x,4.

$$x\left(x-2\right)-8\left(x-2\right)=2x$$

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

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

Use the distributive property to multiply -8 by x-2.

$$x^{2}-2x-8x+16=2x$$

Combine -2x and -8x to get -10x.

$$x^{2}-10x+16=2x$$

Subtract 2x from both sides.

$$x^{2}-10x+16-2x=0$$

Combine -10x and -2x to get -12x.

$$x^{2}-12x+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-12x=-16$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-16+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-16+36$$

Add -16 to 36.

$$x^{2}-12x+36=20$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{20}$$

Simplify.

$$x-6=2\sqrt{5} x-6=-2\sqrt{5}$$

Add 6 to both sides of the equation.

$$x=2\sqrt{5}+6 x=6-2\sqrt{5}$$