Solve x-3/x+6+x-2/x-3=x^2/x^2+3x-18

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

Solve for x

$$x=-1$$

Steps Using Factoring

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 \left(x-3\right)\left(x+6\right), the least common multiple of x+6,x-3,x^{2}+3x-18.

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

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+\left(x+6\right)\left(x-2\right)=x^{2}$$

Use the distributive property to multiply x+6 by x-2 and combine like terms.

$$x^{2}-6x+9+x^{2}+4x-12=x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-6x+9+4x-12=x^{2}$$

Combine -6x and 4x to get -2x.

$$2x^{2}-2x+9-12=x^{2}$$

Subtract 12 from 9 to get -3.

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

Subtract x^{2} from both sides.

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

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

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

To solve the equation, factor x^{2}-2x-3 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-3$$

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. The only such pair is the system solution.

$$a=-3 b=1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-3\right)\left(x+1\right)$$

To find equation solutions, solve x-3=0 and x+1=0.

$$x=3 x=-1$$

Variable x cannot be equal to 3.

$$x=-1$$

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 \left(x-3\right)\left(x+6\right), the least common multiple of x+6,x-3,x^{2}+3x-18.

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

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+\left(x+6\right)\left(x-2\right)=x^{2}$$

Use the distributive property to multiply x+6 by x-2 and combine like terms.

$$x^{2}-6x+9+x^{2}+4x-12=x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-6x+9+4x-12=x^{2}$$

Combine -6x and 4x to get -2x.

$$2x^{2}-2x+9-12=x^{2}$$

Subtract 12 from 9 to get -3.

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

Subtract x^{2} from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-3\right)=-3$$

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. The only such pair is the system solution.

$$a=-3 b=1$$

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

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

Factor out x in x^{2}-3x.

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

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+1\right)$$

To find equation solutions, solve x-3=0 and x+1=0.

$$x=3 x=-1$$

Variable x cannot be equal to 3.

$$x=-1$$

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 \left(x-3\right)\left(x+6\right), the least common multiple of x+6,x-3,x^{2}+3x-18.

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

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+\left(x+6\right)\left(x-2\right)=x^{2}$$

Use the distributive property to multiply x+6 by x-2 and combine like terms.

$$x^{2}-6x+9+x^{2}+4x-12=x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-6x+9+4x-12=x^{2}$$

Combine -6x and 4x to get -2x.

$$2x^{2}-2x+9-12=x^{2}$$

Subtract 12 from 9 to get -3.

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

Subtract x^{2} from both sides.

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

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

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -2.

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

Multiply -4 times -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+12}}{2}$$

Add 4 to 12.

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

Take the square root of 16.

$$x=\frac{-\left(-2\right)±4}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±4}{2}$$

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

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

Divide 6 by 2.

$$x=3$$

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

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=3 x=-1$$

Variable x cannot be equal to 3.

$$x=-1$$

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 \left(x-3\right)\left(x+6\right), the least common multiple of x+6,x-3,x^{2}+3x-18.

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

Multiply x-3 and x-3 to get \left(x-3\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+\left(x+6\right)\left(x-2\right)=x^{2}$$

Use the distributive property to multiply x+6 by x-2 and combine like terms.

$$x^{2}-6x+9+x^{2}+4x-12=x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-6x+9+4x-12=x^{2}$$

Combine -6x and 4x to get -2x.

$$2x^{2}-2x+9-12=x^{2}$$

Subtract 12 from 9 to get -3.

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

Subtract x^{2} from both sides.

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

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

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

Add 3 to both sides. Anything plus zero gives itself.

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

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.

$$x^{2}-2x+1=3+1$$

Add 3 to 1.

$$x^{2}-2x+1=4$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=2 x-1=-2$$

Add 1 to both sides of the equation.

$$x=3 x=-1$$

Variable x cannot be equal to 3.

$$x=-1$$