Solve 1/x-3+18/x^2-9=x/x+3

$$\frac{ 1 }{ x-3 } + \frac{ 18 }{ { x }^{ 2 } -9 } = \frac{ x }{ x+3 }$$

Solve for x

$$x=7$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Add 3 and 18 to get 21.

$$x+21=\left(x-3\right)x$$

Use the distributive property to multiply x-3 by x.

$$x+21=x^{2}-3x$$

Subtract x^{2} from both sides.

$$x+21-x^{2}=-3x$$

Add 3x to both sides.

$$x+21-x^{2}+3x=0$$

Combine x and 3x to get 4x.

$$4x+21-x^{2}=0$$

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

$$-x^{2}+4x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=4 ab=-21=-21$$

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 -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=7 b=-3$$

Rewrite -x^{2}+4x+21 as \left(-x^{2}+7x\right)+\left(-3x+21\right).

$$\left(-x^{2}+7x\right)+\left(-3x+21\right)$$

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

$$-x\left(x-7\right)-3\left(x-7\right)$$

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

$$\left(x-7\right)\left(-x-3\right)$$

To find equation solutions, solve x-7=0 and -x-3=0.

$$x=7 x=-3$$

Variable x cannot be equal to -3.

$$x=7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Add 3 and 18 to get 21.

$$x+21=\left(x-3\right)x$$

Use the distributive property to multiply x-3 by x.

$$x+21=x^{2}-3x$$

Subtract x^{2} from both sides.

$$x+21-x^{2}=-3x$$

Add 3x to both sides.

$$x+21-x^{2}+3x=0$$

Combine x and 3x to get 4x.

$$4x+21-x^{2}=0$$

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.

$$-x^{2}+4x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 21}}{2\left(-1\right)}$$

Multiply 4 times 21.

$$x=\frac{-4±\sqrt{16+84}}{2\left(-1\right)}$$

Add 16 to 84.

$$x=\frac{-4±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-4±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±10}{-2}$$

Now solve the equation x=\frac{-4±10}{-2} when ± is plus. Add -4 to 10.

$$x=\frac{6}{-2}$$

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{-4±10}{-2} when ± is minus. Subtract 10 from -4.

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

Divide -14 by -2.

$$x=7$$

The equation is now solved.

$$x=-3 x=7$$

Variable x cannot be equal to -3.

$$x=7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Add 3 and 18 to get 21.

$$x+21=\left(x-3\right)x$$

Use the distributive property to multiply x-3 by x.

$$x+21=x^{2}-3x$$

Subtract x^{2} from both sides.

$$x+21-x^{2}=-3x$$

Add 3x to both sides.

$$x+21-x^{2}+3x=0$$

Combine x and 3x to get 4x.

$$4x+21-x^{2}=0$$

Subtract 21 from both sides. Anything subtracted from zero gives its negation.

$$4x-x^{2}=-21$$

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.

$$-x^{2}+4x=-21$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{21}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=-\frac{21}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-\frac{21}{-1}$$

Divide -21 by -1.

$$x^{2}-4x=21$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=21+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=21+4$$

Add 21 to 4.

$$x^{2}-4x+4=25$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=5 x-2=-5$$

Add 2 to both sides of the equation.

$$x=7 x=-3$$

Variable x cannot be equal to -3.

$$x=7$$