Solve x/x+6=72/x^2-36+4

$$\frac { x } { x + 6 } = \frac { 72 } { x ^ { 2 } - 36 } + 4$$

Solve for x

$$x=4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -6,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x+6\right), the least common multiple of x+6,x^{2}-36.

$$\left(x-6\right)x=72+\left(x-6\right)\left(x+6\right)\times 4$$

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

$$x^{2}-6x=72+\left(x-6\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x-6 by x+6 and combine like terms.

$$x^{2}-6x=72+\left(x^{2}-36\right)\times 4$$

Use the distributive property to multiply x^{2}-36 by 4.

$$x^{2}-6x=72+4x^{2}-144$$

Subtract 144 from 72 to get -72.

$$x^{2}-6x=-72+4x^{2}$$

Subtract -72 from both sides.

$$x^{2}-6x-\left(-72\right)=4x^{2}$$

The opposite of -72 is 72.

$$x^{2}-6x+72=4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}-6x+72-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+72=0$$

Divide both sides by 3.

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

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.

$$a+b=-2 ab=-24=-24$$

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 -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=4 b=-6$$

Rewrite -x^{2}-2x+24 as \left(-x^{2}+4x\right)+\left(-6x+24\right).

$$\left(-x^{2}+4x\right)+\left(-6x+24\right)$$

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

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

Factor out common term -x+4 by using distributive property.

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

To find equation solutions, solve -x+4=0 and x+6=0.

$$x=4 x=-6$$

Variable x cannot be equal to -6.

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x+6\right), the least common multiple of x+6,x^{2}-36.

$$\left(x-6\right)x=72+\left(x-6\right)\left(x+6\right)\times 4$$

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

$$x^{2}-6x=72+\left(x-6\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x-6 by x+6 and combine like terms.

$$x^{2}-6x=72+\left(x^{2}-36\right)\times 4$$

Use the distributive property to multiply x^{2}-36 by 4.

$$x^{2}-6x=72+4x^{2}-144$$

Subtract 144 from 72 to get -72.

$$x^{2}-6x=-72+4x^{2}$$

Subtract -72 from both sides.

$$x^{2}-6x-\left(-72\right)=4x^{2}$$

The opposite of -72 is 72.

$$x^{2}-6x+72=4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}-6x+72-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -6 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)\times 72}}{2\left(-3\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)\times 72}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+12\times 72}}{2\left(-3\right)}$$

Multiply 12 times 72.

$$x=\frac{-\left(-6\right)±\sqrt{36+864}}{2\left(-3\right)}$$

Add 36 to 864.

$$x=\frac{-\left(-6\right)±\sqrt{900}}{2\left(-3\right)}$$

Take the square root of 900.

$$x=\frac{-\left(-6\right)±30}{2\left(-3\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±30}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{6±30}{-6}$$

Now solve the equation x=\frac{6±30}{-6} when ± is plus. Add 6 to 30.

$$x=\frac{36}{-6}$$

Divide 36 by -6.

$$x=-6$$

Now solve the equation x=\frac{6±30}{-6} when ± is minus. Subtract 30 from 6.

$$x=-\frac{24}{-6}$$

Divide -24 by -6.

$$x=4$$

The equation is now solved.

$$x=-6 x=4$$

Variable x cannot be equal to -6.

$$x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x+6\right), the least common multiple of x+6,x^{2}-36.

$$\left(x-6\right)x=72+\left(x-6\right)\left(x+6\right)\times 4$$

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

$$x^{2}-6x=72+\left(x-6\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x-6 by x+6 and combine like terms.

$$x^{2}-6x=72+\left(x^{2}-36\right)\times 4$$

Use the distributive property to multiply x^{2}-36 by 4.

$$x^{2}-6x=72+4x^{2}-144$$

Subtract 144 from 72 to get -72.

$$x^{2}-6x=-72+4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}-6x-4x^{2}=-72$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x=-72$$

Divide both sides by -3.

$$\frac{-3x^{2}-6x}{-3}=-\frac{72}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{6}{-3}\right)x=-\frac{72}{-3}$$

Divide -6 by -3.

$$x^{2}+2x=-\frac{72}{-3}$$

Divide -72 by -3.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=24+1^{2}$$

Square 1.

$$x^{2}+2x+1=24+1$$

Add 24 to 1.

$$x^{2}+2x+1=25$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+1=5 x+1=-5$$

Subtract 1 from both sides of the equation.

$$x=4 x=-6$$

Variable x cannot be equal to -6.

$$x=4$$