Solve 360/x-360/x+4=3

$$\frac { 360 } { x } - \frac { 360 } { x + 4 } = 3$$

Solve for x

$$x=-24$$

$$x=20$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 360-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 360.

$$360x+1440-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply 3x by x+4.

$$360x+1440-x\times 360=3x^{2}+12x$$

Subtract 3x^{2} from both sides.

$$360x+1440-x\times 360-3x^{2}=12x$$

Subtract 12x from both sides.

$$360x+1440-x\times 360-3x^{2}-12x=0$$

Combine 360x and -12x to get 348x.

$$348x+1440-x\times 360-3x^{2}=0$$

Multiply -1 and 360 to get -360.

$$348x+1440-360x-3x^{2}=0$$

Combine 348x and -360x to get -12x.

$$-12x+1440-3x^{2}=0$$

Divide both sides by 3.

$$-4x+480-x^{2}=0$$

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

$$-x^{2}-4x+480=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+480. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-480=-480$$

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 -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Calculate the sum for each pair.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

The solution is the pair that gives sum -4.

$$a=20 b=-24$$

Rewrite -x^{2}-4x+480 as \left(-x^{2}+20x\right)+\left(-24x+480\right).

$$\left(-x^{2}+20x\right)+\left(-24x+480\right)$$

Factor out x in the first and 24 in the second group.

$$x\left(-x+20\right)+24\left(-x+20\right)$$

Factor out common term -x+20 by using distributive property.

$$\left(-x+20\right)\left(x+24\right)$$

To find equation solutions, solve -x+20=0 and x+24=0.

$$x=20 x=-24$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 360-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 360.

$$360x+1440-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply 3x by x+4.

$$360x+1440-x\times 360=3x^{2}+12x$$

Subtract 3x^{2} from both sides.

$$360x+1440-x\times 360-3x^{2}=12x$$

Subtract 12x from both sides.

$$360x+1440-x\times 360-3x^{2}-12x=0$$

Combine 360x and -12x to get 348x.

$$348x+1440-x\times 360-3x^{2}=0$$

Multiply -1 and 360 to get -360.

$$348x+1440-360x-3x^{2}=0$$

Combine 348x and -360x to get -12x.

$$-12x+1440-3x^{2}=0$$

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.

$$-3x^{2}-12x+1440=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -12 for b, and 1440 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 1440}}{2\left(-3\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 1440}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+12\times 1440}}{2\left(-3\right)}$$

Multiply 12 times 1440.

$$x=\frac{-\left(-12\right)±\sqrt{144+17280}}{2\left(-3\right)}$$

Add 144 to 17280.

$$x=\frac{-\left(-12\right)±\sqrt{17424}}{2\left(-3\right)}$$

Take the square root of 17424.

$$x=\frac{-\left(-12\right)±132}{2\left(-3\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±132}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{12±132}{-6}$$

Now solve the equation x=\frac{12±132}{-6} when ± is plus. Add 12 to 132.

$$x=\frac{144}{-6}$$

Divide 144 by -6.

$$x=-24$$

Now solve the equation x=\frac{12±132}{-6} when ± is minus. Subtract 132 from 12.

$$x=-\frac{120}{-6}$$

Divide -120 by -6.

$$x=20$$

The equation is now solved.

$$x=-24 x=20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 360-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 360.

$$360x+1440-x\times 360=3x\left(x+4\right)$$

Use the distributive property to multiply 3x by x+4.

$$360x+1440-x\times 360=3x^{2}+12x$$

Subtract 3x^{2} from both sides.

$$360x+1440-x\times 360-3x^{2}=12x$$

Subtract 12x from both sides.

$$360x+1440-x\times 360-3x^{2}-12x=0$$

Combine 360x and -12x to get 348x.

$$348x+1440-x\times 360-3x^{2}=0$$

Subtract 1440 from both sides. Anything subtracted from zero gives its negation.

$$348x-x\times 360-3x^{2}=-1440$$

Multiply -1 and 360 to get -360.

$$348x-360x-3x^{2}=-1440$$

Combine 348x and -360x to get -12x.

$$-12x-3x^{2}=-1440$$

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.

$$-3x^{2}-12x=-1440$$

Divide both sides by -3.

$$\frac{-3x^{2}-12x}{-3}=-\frac{1440}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{12}{-3}\right)x=-\frac{1440}{-3}$$

Divide -12 by -3.

$$x^{2}+4x=-\frac{1440}{-3}$$

Divide -1440 by -3.

$$x^{2}+4x=480$$

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.

$$x^{2}+4x+2^{2}=480+2^{2}$$

Square 2.

$$x^{2}+4x+4=480+4$$

Add 480 to 4.

$$x^{2}+4x+4=484$$

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}.

$$\left(x+2\right)^{2}=484$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{484}$$

Simplify.

$$x+2=22 x+2=-22$$

Subtract 2 from both sides of the equation.

$$x=20 x=-24$$