Solve 2x+9/x^2-6x+9-x/2x-6=0

$$\frac { 2 x + 9 } { x ^ { 2 } - 6 x + 9 } - \frac { x } { 2 x - 6 } = 0$$

Solve for x

$$x=-2$$

$$x=9$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)^{2}, the least common multiple of x^{2}-6x+9,2x-6.

$$2\left(2x+9\right)-\left(x-3\right)x=0$$

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

$$4x+18-\left(x-3\right)x=0$$

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

$$4x+18-\left(x^{2}-3x\right)=0$$

To find the opposite of x^{2}-3x, find the opposite of each term.

$$4x+18-x^{2}+3x=0$$

Combine 4x and 3x to get 7x.

$$7x+18-x^{2}=0$$

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

$$-x^{2}+7x+18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=7 ab=-18=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=9 b=-2$$

Rewrite -x^{2}+7x+18 as \left(-x^{2}+9x\right)+\left(-2x+18\right).

$$\left(-x^{2}+9x\right)+\left(-2x+18\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-9\right)-2\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(-x-2\right)$$

To find equation solutions, solve x-9=0 and -x-2=0.

$$x=9 x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)^{2}, the least common multiple of x^{2}-6x+9,2x-6.

$$2\left(2x+9\right)-\left(x-3\right)x=0$$

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

$$4x+18-\left(x-3\right)x=0$$

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

$$4x+18-\left(x^{2}-3x\right)=0$$

To find the opposite of x^{2}-3x, find the opposite of each term.

$$4x+18-x^{2}+3x=0$$

Combine 4x and 3x to get 7x.

$$7x+18-x^{2}=0$$

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.

$$-x^{2}+7x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-7±\sqrt{49+4\times 18}}{2\left(-1\right)}$$

Multiply 4 times 18.

$$x=\frac{-7±\sqrt{49+72}}{2\left(-1\right)}$$

Add 49 to 72.

$$x=\frac{-7±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-7±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-7±11}{-2}$$

Now solve the equation x=\frac{-7±11}{-2} when ± is plus. Add -7 to 11.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-7±11}{-2} when ± is minus. Subtract 11 from -7.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=-2 x=9$$

Steps for Completing the Square

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)^{2}, the least common multiple of x^{2}-6x+9,2x-6.

$$2\left(2x+9\right)-\left(x-3\right)x=0$$

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

$$4x+18-\left(x-3\right)x=0$$

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

$$4x+18-\left(x^{2}-3x\right)=0$$

To find the opposite of x^{2}-3x, find the opposite of each term.

$$4x+18-x^{2}+3x=0$$

Combine 4x and 3x to get 7x.

$$7x+18-x^{2}=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$7x-x^{2}=-18$$

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.

$$-x^{2}+7x=-18$$

Divide both sides by -1.

$$\frac{-x^{2}+7x}{-1}=-\frac{18}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{7}{-1}x=-\frac{18}{-1}$$

Divide 7 by -1.

$$x^{2}-7x=-\frac{18}{-1}$$

Divide -18 by -1.

$$x^{2}-7x=18$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=18+\frac{49}{4}$$

Add 18 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{121}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{11}{2} x-\frac{7}{2}=-\frac{11}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=9 x=-2$$