Solve 54/x+63/x+6=3

$$\frac{ 54 }{ x } + \frac{ 63 }{ x+6 } =3$$

Solve for x

$$x=-3$$

$$x=36$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -6,0 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.

$$\left(x+6\right)\times 54+x\times 63=3x\left(x+6\right)$$

Use the distributive property to multiply x+6 by 54.

$$54x+324+x\times 63=3x\left(x+6\right)$$

Combine 54x and x\times 63 to get 117x.

$$117x+324=3x\left(x+6\right)$$

Use the distributive property to multiply 3x by x+6.

$$117x+324=3x^{2}+18x$$

Subtract 3x^{2} from both sides.

$$117x+324-3x^{2}=18x$$

Subtract 18x from both sides.

$$117x+324-3x^{2}-18x=0$$

Combine 117x and -18x to get 99x.

$$99x+324-3x^{2}=0$$

Divide both sides by 3.

$$33x+108-x^{2}=0$$

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

$$-x^{2}+33x+108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+108. To find a and b, set up a system to be solved.

$$a+b=33 ab=-108=-108$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -108.

$$-1,108 -2,54 -3,36 -4,27 -6,18 -9,12$$

Calculate the sum for each pair.

$$-1+108=107 -2+54=52 -3+36=33 -4+27=23 -6+18=12 -9+12=3$$

The solution is the pair that gives sum 33.

$$a=36 b=-3$$

Rewrite -x^{2}+33x+108 as \left(-x^{2}+36x\right)+\left(-3x+108\right).

$$\left(-x^{2}+36x\right)+\left(-3x+108\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-36\right)-3\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(-x-3\right)$$

To find equation solutions, solve x-36=0 and -x-3=0.

$$x=36 x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,0 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.

$$\left(x+6\right)\times 54+x\times 63=3x\left(x+6\right)$$

Use the distributive property to multiply x+6 by 54.

$$54x+324+x\times 63=3x\left(x+6\right)$$

Combine 54x and x\times 63 to get 117x.

$$117x+324=3x\left(x+6\right)$$

Use the distributive property to multiply 3x by x+6.

$$117x+324=3x^{2}+18x$$

Subtract 3x^{2} from both sides.

$$117x+324-3x^{2}=18x$$

Subtract 18x from both sides.

$$117x+324-3x^{2}-18x=0$$

Combine 117x and -18x to get 99x.

$$99x+324-3x^{2}=0$$

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.

$$-3x^{2}+99x+324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 99 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-99±\sqrt{99^{2}-4\left(-3\right)\times 324}}{2\left(-3\right)}$$

Square 99.

$$x=\frac{-99±\sqrt{9801-4\left(-3\right)\times 324}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-99±\sqrt{9801+12\times 324}}{2\left(-3\right)}$$

Multiply 12 times 324.

$$x=\frac{-99±\sqrt{9801+3888}}{2\left(-3\right)}$$

Add 9801 to 3888.

$$x=\frac{-99±\sqrt{13689}}{2\left(-3\right)}$$

Take the square root of 13689.

$$x=\frac{-99±117}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-99±117}{-6}$$

Now solve the equation x=\frac{-99±117}{-6} when ± is plus. Add -99 to 117.

$$x=\frac{18}{-6}$$

Divide 18 by -6.

$$x=-3$$

Now solve the equation x=\frac{-99±117}{-6} when ± is minus. Subtract 117 from -99.

$$x=-\frac{216}{-6}$$

Divide -216 by -6.

$$x=36$$

The equation is now solved.

$$x=-3 x=36$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,0 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.

$$\left(x+6\right)\times 54+x\times 63=3x\left(x+6\right)$$

Use the distributive property to multiply x+6 by 54.

$$54x+324+x\times 63=3x\left(x+6\right)$$

Combine 54x and x\times 63 to get 117x.

$$117x+324=3x\left(x+6\right)$$

Use the distributive property to multiply 3x by x+6.

$$117x+324=3x^{2}+18x$$

Subtract 3x^{2} from both sides.

$$117x+324-3x^{2}=18x$$

Subtract 18x from both sides.

$$117x+324-3x^{2}-18x=0$$

Combine 117x and -18x to get 99x.

$$99x+324-3x^{2}=0$$

Subtract 324 from both sides. Anything subtracted from zero gives its negation.

$$99x-3x^{2}=-324$$

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.

$$-3x^{2}+99x=-324$$

Divide both sides by -3.

$$\frac{-3x^{2}+99x}{-3}=-\frac{324}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{99}{-3}x=-\frac{324}{-3}$$

Divide 99 by -3.

$$x^{2}-33x=-\frac{324}{-3}$$

Divide -324 by -3.

$$x^{2}-33x=108$$

Divide -33, the coefficient of the x term, by 2 to get -\frac{33}{2}. Then add the square of -\frac{33}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-33x+\left(-\frac{33}{2}\right)^{2}=108+\left(-\frac{33}{2}\right)^{2}$$

Square -\frac{33}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-33x+\frac{1089}{4}=108+\frac{1089}{4}$$

Add 108 to \frac{1089}{4}.

$$x^{2}-33x+\frac{1089}{4}=\frac{1521}{4}$$

Factor x^{2}-33x+\frac{1089}{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}.

$$\left(x-\frac{33}{2}\right)^{2}=\frac{1521}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{2}\right)^{2}}=\sqrt{\frac{1521}{4}}$$

Simplify.

$$x-\frac{33}{2}=\frac{39}{2} x-\frac{33}{2}=-\frac{39}{2}$$

Add \frac{33}{2} to both sides of the equation.

$$x=36 x=-3$$