Solve 8/x-x/4=1

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

Solve for x

$$x=-8$$

$$x=4$$

Steps Using Factoring By Grouping

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 x,4.

$$4\times 8-xx=4x$$

Multiply x and x to get x^{2}.

$$4\times 8-x^{2}=4x$$

Multiply 4 and 8 to get 32.

$$32-x^{2}=4x$$

Subtract 4x from both sides.

$$32-x^{2}-4x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-4x+32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+32. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-32=-32$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -32.

$$1,-32 2,-16 4,-8$$

Calculate the sum for each pair.

$$1-32=-31 2-16=-14 4-8=-4$$

The solution is the pair that gives sum -4.

$$a=4 b=-8$$

Rewrite -x^{2}-4x+32 as \left(-x^{2}+4x\right)+\left(-8x+32\right).

$$\left(-x^{2}+4x\right)+\left(-8x+32\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x+4\right)+8\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+8\right)$$

To find equation solutions, solve -x+4=0 and x+8=0.

$$x=4 x=-8$$

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 x,4.

$$4\times 8-xx=4x$$

Multiply x and x to get x^{2}.

$$4\times 8-x^{2}=4x$$

Multiply 4 and 8 to get 32.

$$32-x^{2}=4x$$

Subtract 4x from both sides.

$$32-x^{2}-4x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-4x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 32}}{2\left(-1\right)}$$

Multiply 4 times 32.

$$x=\frac{-\left(-4\right)±\sqrt{16+128}}{2\left(-1\right)}$$

Add 16 to 128.

$$x=\frac{-\left(-4\right)±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-\left(-4\right)±12}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±12}{-2}$$

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

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

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

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

Divide -8 by -2.

$$x=4$$

The equation is now solved.

$$x=-8 x=4$$

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 x,4.

$$4\times 8-xx=4x$$

Multiply x and x to get x^{2}.

$$4\times 8-x^{2}=4x$$

Multiply 4 and 8 to get 32.

$$32-x^{2}=4x$$

Subtract 4x from both sides.

$$32-x^{2}-4x=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}-4x=-32$$

Divide both sides by -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{32}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{32}{-1}$$

Divide -4 by -1.

$$x^{2}+4x=-\frac{32}{-1}$$

Divide -32 by -1.

$$x^{2}+4x=32$$

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.

$$x^{2}+4x+2^{2}=32+2^{2}$$

Square 2.

$$x^{2}+4x+4=32+4$$

Add 32 to 4.

$$x^{2}+4x+4=36$$

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}.

$$\left(x+2\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+2=6 x+2=-6$$

Subtract 2 from both sides of the equation.

$$x=4 x=-8$$