Solve 18/x-4/20-x=1

$$\frac { 18 } { x } - \frac { 4 } { 20 - x } = 1$$

Solve for x

$$x=12$$

$$x=30$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,20 since division by zero is not defined. Multiply both sides of the equation by x\left(x-20\right), the least common multiple of x,20-x.

$$\left(x-20\right)\times 18-\left(-x\times 4\right)=x\left(x-20\right)$$

Use the distributive property to multiply x-20 by 18.

$$18x-360-\left(-x\times 4\right)=x\left(x-20\right)$$

Multiply -1 and 4 to get -4.

$$18x-360-\left(-4x\right)=x\left(x-20\right)$$

The opposite of -4x is 4x.

$$18x-360+4x=x\left(x-20\right)$$

Combine 18x and 4x to get 22x.

$$22x-360=x\left(x-20\right)$$

Use the distributive property to multiply x by x-20.

$$22x-360=x^{2}-20x$$

Subtract x^{2} from both sides.

$$22x-360-x^{2}=-20x$$

Add 20x to both sides.

$$22x-360-x^{2}+20x=0$$

Combine 22x and 20x to get 42x.

$$42x-360-x^{2}=0$$

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

$$-x^{2}+42x-360=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-360. To find a and b, set up a system to be solved.

$$a+b=42 ab=-\left(-360\right)=360$$

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 360.

$$1,360 2,180 3,120 4,90 5,72 6,60 8,45 9,40 10,36 12,30 15,24 18,20$$

Calculate the sum for each pair.

$$1+360=361 2+180=182 3+120=123 4+90=94 5+72=77 6+60=66 8+45=53 9+40=49 10+36=46 12+30=42 15+24=39 18+20=38$$

The solution is the pair that gives sum 42.

$$a=30 b=12$$

Rewrite -x^{2}+42x-360 as \left(-x^{2}+30x\right)+\left(12x-360\right).

$$\left(-x^{2}+30x\right)+\left(12x-360\right)$$

Factor out -x in the first and 12 in the second group.

$$-x\left(x-30\right)+12\left(x-30\right)$$

Factor out common term x-30 by using distributive property.

$$\left(x-30\right)\left(-x+12\right)$$

To find equation solutions, solve x-30=0 and -x+12=0.

$$x=30 x=12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,20 since division by zero is not defined. Multiply both sides of the equation by x\left(x-20\right), the least common multiple of x,20-x.

$$\left(x-20\right)\times 18-\left(-x\times 4\right)=x\left(x-20\right)$$

Use the distributive property to multiply x-20 by 18.

$$18x-360-\left(-x\times 4\right)=x\left(x-20\right)$$

Multiply -1 and 4 to get -4.

$$18x-360-\left(-4x\right)=x\left(x-20\right)$$

The opposite of -4x is 4x.

$$18x-360+4x=x\left(x-20\right)$$

Combine 18x and 4x to get 22x.

$$22x-360=x\left(x-20\right)$$

Use the distributive property to multiply x by x-20.

$$22x-360=x^{2}-20x$$

Subtract x^{2} from both sides.

$$22x-360-x^{2}=-20x$$

Add 20x to both sides.

$$22x-360-x^{2}+20x=0$$

Combine 22x and 20x to get 42x.

$$42x-360-x^{2}=0$$

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.

$$-x^{2}+42x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 42 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-42±\sqrt{42^{2}-4\left(-1\right)\left(-360\right)}}{2\left(-1\right)}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\left(-1\right)\left(-360\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-42±\sqrt{1764+4\left(-360\right)}}{2\left(-1\right)}$$

Multiply 4 times -360.

$$x=\frac{-42±\sqrt{1764-1440}}{2\left(-1\right)}$$

Add 1764 to -1440.

$$x=\frac{-42±\sqrt{324}}{2\left(-1\right)}$$

Take the square root of 324.

$$x=\frac{-42±18}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-42±18}{-2}$$

Now solve the equation x=\frac{-42±18}{-2} when ± is plus. Add -42 to 18.

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

Divide -24 by -2.

$$x=12$$

Now solve the equation x=\frac{-42±18}{-2} when ± is minus. Subtract 18 from -42.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

The equation is now solved.

$$x=12 x=30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,20 since division by zero is not defined. Multiply both sides of the equation by x\left(x-20\right), the least common multiple of x,20-x.

$$\left(x-20\right)\times 18-\left(-x\times 4\right)=x\left(x-20\right)$$

Use the distributive property to multiply x-20 by 18.

$$18x-360-\left(-x\times 4\right)=x\left(x-20\right)$$

Multiply -1 and 4 to get -4.

$$18x-360-\left(-4x\right)=x\left(x-20\right)$$

The opposite of -4x is 4x.

$$18x-360+4x=x\left(x-20\right)$$

Combine 18x and 4x to get 22x.

$$22x-360=x\left(x-20\right)$$

Use the distributive property to multiply x by x-20.

$$22x-360=x^{2}-20x$$

Subtract x^{2} from both sides.

$$22x-360-x^{2}=-20x$$

Add 20x to both sides.

$$22x-360-x^{2}+20x=0$$

Combine 22x and 20x to get 42x.

$$42x-360-x^{2}=0$$

Add 360 to both sides. Anything plus zero gives itself.

$$42x-x^{2}=360$$

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.

$$-x^{2}+42x=360$$

Divide both sides by -1.

$$\frac{-x^{2}+42x}{-1}=\frac{360}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{42}{-1}x=\frac{360}{-1}$$

Divide 42 by -1.

$$x^{2}-42x=\frac{360}{-1}$$

Divide 360 by -1.

$$x^{2}-42x=-360$$

Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-42x+\left(-21\right)^{2}=-360+\left(-21\right)^{2}$$

Square -21.

$$x^{2}-42x+441=-360+441$$

Add -360 to 441.

$$x^{2}-42x+441=81$$

Factor x^{2}-42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-21\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-21\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-21=9 x-21=-9$$

Add 21 to both sides of the equation.

$$x=30 x=12$$