Solve 4/x+4/x-6=1

$$\frac { 4 } { x } + \frac { 4 } { x - 6 } = 1$$

Solve for x

$$x=2$$

$$x=12$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by x\left(x-6\right), the least common multiple of x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Use the distributive property to multiply x-6 by 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Combine 4x and x\times 4 to get 8x.

$$8x-24=x\left(x-6\right)$$

Use the distributive property to multiply x by x-6.

$$8x-24=x^{2}-6x$$

Subtract x^{2} from both sides.

$$8x-24-x^{2}=-6x$$

Add 6x to both sides.

$$8x-24-x^{2}+6x=0$$

Combine 8x and 6x to get 14x.

$$14x-24-x^{2}=0$$

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

$$-x^{2}+14x-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-24\right)=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 14.

$$a=12 b=2$$

Rewrite -x^{2}+14x-24 as \left(-x^{2}+12x\right)+\left(2x-24\right).

$$\left(-x^{2}+12x\right)+\left(2x-24\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-12\right)+2\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x+2\right)$$

To find equation solutions, solve x-12=0 and -x+2=0.

$$x=12 x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by x\left(x-6\right), the least common multiple of x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Use the distributive property to multiply x-6 by 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Combine 4x and x\times 4 to get 8x.

$$8x-24=x\left(x-6\right)$$

Use the distributive property to multiply x by x-6.

$$8x-24=x^{2}-6x$$

Subtract x^{2} from both sides.

$$8x-24-x^{2}=-6x$$

Add 6x to both sides.

$$8x-24-x^{2}+6x=0$$

Combine 8x and 6x to get 14x.

$$14x-24-x^{2}=0$$

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.

$$-x^{2}+14x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\left(-24\right)}}{2\left(-1\right)}$$

Multiply 4 times -24.

$$x=\frac{-14±\sqrt{196-96}}{2\left(-1\right)}$$

Add 196 to -96.

$$x=\frac{-14±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-14±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14±10}{-2}$$

Now solve the equation x=\frac{-14±10}{-2} when ± is plus. Add -14 to 10.

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

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-14±10}{-2} when ± is minus. Subtract 10 from -14.

$$x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=2 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by x\left(x-6\right), the least common multiple of x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Use the distributive property to multiply x-6 by 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Combine 4x and x\times 4 to get 8x.

$$8x-24=x\left(x-6\right)$$

Use the distributive property to multiply x by x-6.

$$8x-24=x^{2}-6x$$

Subtract x^{2} from both sides.

$$8x-24-x^{2}=-6x$$

Add 6x to both sides.

$$8x-24-x^{2}+6x=0$$

Combine 8x and 6x to get 14x.

$$14x-24-x^{2}=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$14x-x^{2}=24$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+14x=24$$

Divide both sides by -1.

$$\frac{-x^{2}+14x}{-1}=\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{14}{-1}x=\frac{24}{-1}$$

Divide 14 by -1.

$$x^{2}-14x=\frac{24}{-1}$$

Divide 24 by -1.

$$x^{2}-14x=-24$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-24+49$$

Add -24 to 49.

$$x^{2}-14x+49=25$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-7=5 x-7=-5$$

Add 7 to both sides of the equation.

$$x=12 x=2$$