Solve x^2+9/x^2+3x-18-3/x-3+27/x+6=5/2

$$\frac { x ^ { 2 } + 9 } { x ^ { 2 } + 3 x - 18 } - \frac { 3 } { x - 3 } + \frac { 27 } { x + 6 } = \frac { 5 } { 2 }$$

Solve for x

$$x=5$$

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x+6\right), the least common multiple of x^{2}+3x-18,x-3,x+6,2.

$$2\left(x^{2}+9\right)-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2 by x^{2}+9.

$$2x^{2}+18-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+12 by 3.

$$2x^{2}+18-\left(6x+36\right)+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

To find the opposite of 6x+36, find the opposite of each term.

$$2x^{2}+18-6x-36+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Subtract 36 from 18 to get -18.

$$2x^{2}-18-6x+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x-6 by 27.

$$2x^{2}-18-6x+54x-162=5\left(x-3\right)\left(x+6\right)$$

Combine -6x and 54x to get 48x.

$$2x^{2}-18+48x-162=5\left(x-3\right)\left(x+6\right)$$

Subtract 162 from -18 to get -180.

$$2x^{2}-180+48x=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 5 by x-3.

$$2x^{2}-180+48x=\left(5x-15\right)\left(x+6\right)$$

Use the distributive property to multiply 5x-15 by x+6 and combine like terms.

$$2x^{2}-180+48x=5x^{2}+15x-90$$

Subtract 5x^{2} from both sides.

$$2x^{2}-180+48x-5x^{2}=15x-90$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-180+48x=15x-90$$

Subtract 15x from both sides.

$$-3x^{2}-180+48x-15x=-90$$

Combine 48x and -15x to get 33x.

$$-3x^{2}-180+33x=-90$$

Add 90 to both sides.

$$-3x^{2}-180+33x+90=0$$

Add -180 and 90 to get -90.

$$-3x^{2}-90+33x=0$$

Divide both sides by 3.

$$-x^{2}-30+11x=0$$

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

$$-x^{2}+11x-30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=11 ab=-\left(-30\right)=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 11.

$$a=6 b=5$$

Rewrite -x^{2}+11x-30 as \left(-x^{2}+6x\right)+\left(5x-30\right).

$$\left(-x^{2}+6x\right)+\left(5x-30\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-6\right)+5\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x+5\right)$$

To find equation solutions, solve x-6=0 and -x+5=0.

$$x=6 x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x+6\right), the least common multiple of x^{2}+3x-18,x-3,x+6,2.

$$2\left(x^{2}+9\right)-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2 by x^{2}+9.

$$2x^{2}+18-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+12 by 3.

$$2x^{2}+18-\left(6x+36\right)+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

To find the opposite of 6x+36, find the opposite of each term.

$$2x^{2}+18-6x-36+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Subtract 36 from 18 to get -18.

$$2x^{2}-18-6x+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x-6 by 27.

$$2x^{2}-18-6x+54x-162=5\left(x-3\right)\left(x+6\right)$$

Combine -6x and 54x to get 48x.

$$2x^{2}-18+48x-162=5\left(x-3\right)\left(x+6\right)$$

Subtract 162 from -18 to get -180.

$$2x^{2}-180+48x=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 5 by x-3.

$$2x^{2}-180+48x=\left(5x-15\right)\left(x+6\right)$$

Use the distributive property to multiply 5x-15 by x+6 and combine like terms.

$$2x^{2}-180+48x=5x^{2}+15x-90$$

Subtract 5x^{2} from both sides.

$$2x^{2}-180+48x-5x^{2}=15x-90$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-180+48x=15x-90$$

Subtract 15x from both sides.

$$-3x^{2}-180+48x-15x=-90$$

Combine 48x and -15x to get 33x.

$$-3x^{2}-180+33x=-90$$

Add 90 to both sides.

$$-3x^{2}-180+33x+90=0$$

Add -180 and 90 to get -90.

$$-3x^{2}-90+33x=0$$

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.

$$-3x^{2}+33x-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 33 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-3\right)\left(-90\right)}}{2\left(-3\right)}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-3\right)\left(-90\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-33±\sqrt{1089+12\left(-90\right)}}{2\left(-3\right)}$$

Multiply 12 times -90.

$$x=\frac{-33±\sqrt{1089-1080}}{2\left(-3\right)}$$

Add 1089 to -1080.

$$x=\frac{-33±\sqrt{9}}{2\left(-3\right)}$$

Take the square root of 9.

$$x=\frac{-33±3}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-33±3}{-6}$$

Now solve the equation x=\frac{-33±3}{-6} when ± is plus. Add -33 to 3.

$$x=-\frac{30}{-6}$$

Divide -30 by -6.

$$x=5$$

Now solve the equation x=\frac{-33±3}{-6} when ± is minus. Subtract 3 from -33.

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

Divide -36 by -6.

$$x=6$$

The equation is now solved.

$$x=5 x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x+6\right), the least common multiple of x^{2}+3x-18,x-3,x+6,2.

$$2\left(x^{2}+9\right)-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2 by x^{2}+9.

$$2x^{2}+18-\left(2x+12\right)\times 3+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+12 by 3.

$$2x^{2}+18-\left(6x+36\right)+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

To find the opposite of 6x+36, find the opposite of each term.

$$2x^{2}+18-6x-36+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Subtract 36 from 18 to get -18.

$$2x^{2}-18-6x+\left(2x-6\right)\times 27=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 2x-6 by 27.

$$2x^{2}-18-6x+54x-162=5\left(x-3\right)\left(x+6\right)$$

Combine -6x and 54x to get 48x.

$$2x^{2}-18+48x-162=5\left(x-3\right)\left(x+6\right)$$

Subtract 162 from -18 to get -180.

$$2x^{2}-180+48x=5\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 5 by x-3.

$$2x^{2}-180+48x=\left(5x-15\right)\left(x+6\right)$$

Use the distributive property to multiply 5x-15 by x+6 and combine like terms.

$$2x^{2}-180+48x=5x^{2}+15x-90$$

Subtract 5x^{2} from both sides.

$$2x^{2}-180+48x-5x^{2}=15x-90$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-180+48x=15x-90$$

Subtract 15x from both sides.

$$-3x^{2}-180+48x-15x=-90$$

Combine 48x and -15x to get 33x.

$$-3x^{2}-180+33x=-90$$

Add 180 to both sides.

$$-3x^{2}+33x=-90+180$$

Add -90 and 180 to get 90.

$$-3x^{2}+33x=90$$

Divide both sides by -3.

$$\frac{-3x^{2}+33x}{-3}=\frac{90}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{33}{-3}x=\frac{90}{-3}$$

Divide 33 by -3.

$$x^{2}-11x=\frac{90}{-3}$$

Divide 90 by -3.

$$x^{2}-11x=-30$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-30+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-30+\frac{121}{4}$$

Add -30 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{1}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{1}{2} x-\frac{11}{2}=-\frac{1}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=6 x=5$$