Solve 1/x-6+7-x/x+6=13/x^2-36

$$\frac { 1 } { x - 6 } + \frac { 7 - x } { x + 6 } = \frac { 13 } { x ^ { 2 } - 36 }$$

Solve for x

$$x=7$$

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+6,x^{2}-36.

$$x+6+\left(x-6\right)\left(7-x\right)=13$$

Use the distributive property to multiply x-6 by 7-x and combine like terms.

$$x+6+13x-x^{2}-42=13$$

Combine x and 13x to get 14x.

$$14x+6-x^{2}-42=13$$

Subtract 42 from 6 to get -36.

$$14x-36-x^{2}=13$$

Subtract 13 from both sides.

$$14x-36-x^{2}-13=0$$

Subtract 13 from -36 to get -49.

$$14x-49-x^{2}=0$$

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

$$-x^{2}+14x-49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-49. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-49\right)=49$$

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 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 14.

$$a=7 b=7$$

Rewrite -x^{2}+14x-49 as \left(-x^{2}+7x\right)+\left(7x-49\right).

$$\left(-x^{2}+7x\right)+\left(7x-49\right)$$

Factor out -x in the first and 7 in the second group.

$$-x\left(x-7\right)+7\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x+7\right)$$

To find equation solutions, solve x-7=0 and -x+7=0.

$$x=7 x=7$$

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+6,x^{2}-36.

$$x+6+\left(x-6\right)\left(7-x\right)=13$$

Use the distributive property to multiply x-6 by 7-x and combine like terms.

$$x+6+13x-x^{2}-42=13$$

Combine x and 13x to get 14x.

$$14x+6-x^{2}-42=13$$

Subtract 42 from 6 to get -36.

$$14x-36-x^{2}=13$$

Subtract 13 from both sides.

$$14x-36-x^{2}-13=0$$

Subtract 13 from -36 to get -49.

$$14x-49-x^{2}=0$$

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.

$$-x^{2}+14x-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\left(-49\right)}}{2\left(-1\right)}$$

Multiply 4 times -49.

$$x=\frac{-14±\sqrt{196-196}}{2\left(-1\right)}$$

Add 196 to -196.

$$x=\frac{-14±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

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+6,x^{2}-36.

$$x+6+\left(x-6\right)\left(7-x\right)=13$$

Use the distributive property to multiply x-6 by 7-x and combine like terms.

$$x+6+13x-x^{2}-42=13$$

Combine x and 13x to get 14x.

$$14x+6-x^{2}-42=13$$

Subtract 42 from 6 to get -36.

$$14x-36-x^{2}=13$$

Add 36 to both sides.

$$14x-x^{2}=13+36$$

Add 13 and 36 to get 49.

$$14x-x^{2}=49$$

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.

$$-x^{2}+14x=49$$

Divide both sides by -1.

$$\frac{-x^{2}+14x}{-1}=\frac{49}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{14}{-1}x=\frac{49}{-1}$$

Divide 14 by -1.

$$x^{2}-14x=\frac{49}{-1}$$

Divide 49 by -1.

$$x^{2}-14x=-49$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-49+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-49+49$$

Add -49 to 49.

$$x^{2}-14x+49=0$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-7=0 x-7=0$$

Add 7 to both sides of the equation.

$$x=7 x=7$$

The equation is now solved. Solutions are the same.

$$x=7$$