Solve 8/(frac{12{x+12})(12/x+12)x}=3

$$\frac{ 8 }{ ( \frac{ 12 }{ x+12 } )( \frac{ 12 }{ x+12 } )x } =3$$

Solve for x

$$x=6$$

$$x=24$$

Steps Using Factoring

Multiply \frac{12}{x+12} and \frac{12}{x+12} to get \left(\frac{12}{x+12}\right)^{2}.

$$\frac{8}{\left(\frac{12}{x+12}\right)^{2}x}=3$$

To raise \frac{12}{x+12} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{\frac{12^{2}}{\left(x+12\right)^{2}}x}=3$$

Express \frac{12^{2}}{\left(x+12\right)^{2}}x as a single fraction.

$$\frac{8}{\frac{12^{2}x}{\left(x+12\right)^{2}}}=3$$

Variable x cannot be equal to -12 since division by zero is not defined. Divide 8 by \frac{12^{2}x}{\left(x+12\right)^{2}} by multiplying 8 by the reciprocal of \frac{12^{2}x}{\left(x+12\right)^{2}}.

$$\frac{8\left(x+12\right)^{2}}{12^{2}x}=3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$\frac{8\left(x^{2}+24x+144\right)}{12^{2}x}=3$$

Calculate 12 to the power of 2 and get 144.

$$\frac{8\left(x^{2}+24x+144\right)}{144x}=3$$

Cancel out 8 in both numerator and denominator.

$$\frac{x^{2}+24x+144}{18x}=3$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 18x.

$$x^{2}+24x+144=54x$$

Subtract 54x from both sides.

$$x^{2}+24x+144-54x=0$$

Combine 24x and -54x to get -30x.

$$x^{2}-30x+144=0$$

To solve the equation, factor x^{2}-30x+144 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-30 ab=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -30.

$$a=-24 b=-6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

To find equation solutions, solve x-24=0 and x-6=0.

$$x=24 x=6$$

Steps Using Factoring By Grouping

Multiply \frac{12}{x+12} and \frac{12}{x+12} to get \left(\frac{12}{x+12}\right)^{2}.

$$\frac{8}{\left(\frac{12}{x+12}\right)^{2}x}=3$$

To raise \frac{12}{x+12} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{\frac{12^{2}}{\left(x+12\right)^{2}}x}=3$$

Express \frac{12^{2}}{\left(x+12\right)^{2}}x as a single fraction.

$$\frac{8}{\frac{12^{2}x}{\left(x+12\right)^{2}}}=3$$

Variable x cannot be equal to -12 since division by zero is not defined. Divide 8 by \frac{12^{2}x}{\left(x+12\right)^{2}} by multiplying 8 by the reciprocal of \frac{12^{2}x}{\left(x+12\right)^{2}}.

$$\frac{8\left(x+12\right)^{2}}{12^{2}x}=3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$\frac{8\left(x^{2}+24x+144\right)}{12^{2}x}=3$$

Calculate 12 to the power of 2 and get 144.

$$\frac{8\left(x^{2}+24x+144\right)}{144x}=3$$

Cancel out 8 in both numerator and denominator.

$$\frac{x^{2}+24x+144}{18x}=3$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 18x.

$$x^{2}+24x+144=54x$$

Subtract 54x from both sides.

$$x^{2}+24x+144-54x=0$$

Combine 24x and -54x to get -30x.

$$x^{2}-30x+144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+144. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\times 144=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -30.

$$a=-24 b=-6$$

Rewrite x^{2}-30x+144 as \left(x^{2}-24x\right)+\left(-6x+144\right).

$$\left(x^{2}-24x\right)+\left(-6x+144\right)$$

Factor out x in the first and -6 in the second group.

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

Factor out common term x-24 by using distributive property.

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

To find equation solutions, solve x-24=0 and x-6=0.

$$x=24 x=6$$

Steps Using the Quadratic Formula

Multiply \frac{12}{x+12} and \frac{12}{x+12} to get \left(\frac{12}{x+12}\right)^{2}.

$$\frac{8}{\left(\frac{12}{x+12}\right)^{2}x}=3$$

To raise \frac{12}{x+12} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{\frac{12^{2}}{\left(x+12\right)^{2}}x}=3$$

Express \frac{12^{2}}{\left(x+12\right)^{2}}x as a single fraction.

$$\frac{8}{\frac{12^{2}x}{\left(x+12\right)^{2}}}=3$$

Variable x cannot be equal to -12 since division by zero is not defined. Divide 8 by \frac{12^{2}x}{\left(x+12\right)^{2}} by multiplying 8 by the reciprocal of \frac{12^{2}x}{\left(x+12\right)^{2}}.

$$\frac{8\left(x+12\right)^{2}}{12^{2}x}=3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$\frac{8\left(x^{2}+24x+144\right)}{12^{2}x}=3$$

Calculate 12 to the power of 2 and get 144.

$$\frac{8\left(x^{2}+24x+144\right)}{144x}=3$$

Cancel out 8 in both numerator and denominator.

$$\frac{x^{2}+24x+144}{18x}=3$$

Subtract 3 from both sides.

$$\frac{x^{2}+24x+144}{18x}-3=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{18x}{18x}.

$$\frac{x^{2}+24x+144}{18x}-\frac{3\times 18x}{18x}=0$$

Since \frac{x^{2}+24x+144}{18x} and \frac{3\times 18x}{18x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+24x+144-3\times 18x}{18x}=0$$

Do the multiplications in x^{2}+24x+144-3\times 18x.

$$\frac{x^{2}+24x+144-54x}{18x}=0$$

Combine like terms in x^{2}+24x+144-54x.

$$\frac{x^{2}-30x+144}{18x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 18x.

$$x^{2}-30x+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 144}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 144}}{2}$$

Multiply -4 times 144.

$$x=\frac{-\left(-30\right)±\sqrt{900-576}}{2}$$

Add 900 to -576.

$$x=\frac{-\left(-30\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-30\right)±18}{2}$$

The opposite of -30 is 30.

$$x=\frac{30±18}{2}$$

Now solve the equation x=\frac{30±18}{2} when ± is plus. Add 30 to 18.

$$x=\frac{48}{2}$$

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{30±18}{2} when ± is minus. Subtract 18 from 30.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

The equation is now solved.

$$x=24 x=6$$

Steps for Completing the Square

Multiply \frac{12}{x+12} and \frac{12}{x+12} to get \left(\frac{12}{x+12}\right)^{2}.

$$\frac{8}{\left(\frac{12}{x+12}\right)^{2}x}=3$$

To raise \frac{12}{x+12} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{\frac{12^{2}}{\left(x+12\right)^{2}}x}=3$$

Express \frac{12^{2}}{\left(x+12\right)^{2}}x as a single fraction.

$$\frac{8}{\frac{12^{2}x}{\left(x+12\right)^{2}}}=3$$

Variable x cannot be equal to -12 since division by zero is not defined. Divide 8 by \frac{12^{2}x}{\left(x+12\right)^{2}} by multiplying 8 by the reciprocal of \frac{12^{2}x}{\left(x+12\right)^{2}}.

$$\frac{8\left(x+12\right)^{2}}{12^{2}x}=3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$\frac{8\left(x^{2}+24x+144\right)}{12^{2}x}=3$$

Calculate 12 to the power of 2 and get 144.

$$\frac{8\left(x^{2}+24x+144\right)}{144x}=3$$

Cancel out 8 in both numerator and denominator.

$$\frac{x^{2}+24x+144}{18x}=3$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 18x.

$$x^{2}+24x+144=54x$$

Subtract 54x from both sides.

$$x^{2}+24x+144-54x=0$$

Combine 24x and -54x to get -30x.

$$x^{2}-30x+144=0$$

Subtract 144 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-30x=-144$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-144+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-144+225$$

Add -144 to 225.

$$x^{2}-30x+225=81$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-15=9 x-15=-9$$

Add 15 to both sides of the equation.

$$x=24 x=6$$