Solve 2x+1/x^2+x-6-x-1/x^2-5x+6=6/x^2-9

$$\frac { 2 x + 1 } { x ^ { 2 } + x - 6 } - \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } = \frac { 6 } { x ^ { 2 } - 9 }$$

Solve for x

$$x=1$$

$$x=12$$

Steps Using Factoring

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}+x-6,x^{2}-5x+6,x^{2}-9.

$$\left(x-3\right)\left(2x+1\right)-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x-3 by 2x+1 and combine like terms.

$$2x^{2}-5x-3-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x+3 by x-1 and combine like terms.

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

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

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

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

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

Combine -5x and -2x to get -7x.

$$x^{2}-7x-3+3=\left(x-2\right)\times 6$$

Add -3 and 3 to get 0.

$$x^{2}-7x=\left(x-2\right)\times 6$$

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

$$x^{2}-7x=6x-12$$

Subtract 6x from both sides.

$$x^{2}-7x-6x=-12$$

Combine -7x and -6x to get -13x.

$$x^{2}-13x=-12$$

Add 12 to both sides.

$$x^{2}-13x+12=0$$

To solve the equation, factor x^{2}-13x+12 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.

$$a+b=-13 ab=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -13.

$$a=-12 b=-1$$

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

$$\left(x-12\right)\left(x-1\right)$$

To find equation solutions, solve x-12=0 and x-1=0.

$$x=12 x=1$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}+x-6,x^{2}-5x+6,x^{2}-9.

$$\left(x-3\right)\left(2x+1\right)-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x-3 by 2x+1 and combine like terms.

$$2x^{2}-5x-3-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x+3 by x-1 and combine like terms.

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

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

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

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

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

Combine -5x and -2x to get -7x.

$$x^{2}-7x-3+3=\left(x-2\right)\times 6$$

Add -3 and 3 to get 0.

$$x^{2}-7x=\left(x-2\right)\times 6$$

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

$$x^{2}-7x=6x-12$$

Subtract 6x from both sides.

$$x^{2}-7x-6x=-12$$

Combine -7x and -6x to get -13x.

$$x^{2}-13x=-12$$

Add 12 to both sides.

$$x^{2}-13x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\times 12=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -13.

$$a=-12 b=-1$$

Rewrite x^{2}-13x+12 as \left(x^{2}-12x\right)+\left(-x+12\right).

$$\left(x^{2}-12x\right)+\left(-x+12\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-12\right)-\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x-1\right)$$

To find equation solutions, solve x-12=0 and x-1=0.

$$x=12 x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}+x-6,x^{2}-5x+6,x^{2}-9.

$$\left(x-3\right)\left(2x+1\right)-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x-3 by 2x+1 and combine like terms.

$$2x^{2}-5x-3-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x+3 by x-1 and combine like terms.

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

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

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

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

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

Combine -5x and -2x to get -7x.

$$x^{2}-7x-3+3=\left(x-2\right)\times 6$$

Add -3 and 3 to get 0.

$$x^{2}-7x=\left(x-2\right)\times 6$$

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

$$x^{2}-7x=6x-12$$

Subtract 6x from both sides.

$$x^{2}-7x-6x=-12$$

Combine -7x and -6x to get -13x.

$$x^{2}-13x=-12$$

Add 12 to both sides.

$$x^{2}-13x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 12}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-13\right)±\sqrt{169-48}}{2}$$

Add 169 to -48.

$$x=\frac{-\left(-13\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-\left(-13\right)±11}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±11}{2}$$

Now solve the equation x=\frac{13±11}{2} when ± is plus. Add 13 to 11.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{13±11}{2} when ± is minus. Subtract 11 from 13.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=12 x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}+x-6,x^{2}-5x+6,x^{2}-9.

$$\left(x-3\right)\left(2x+1\right)-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x-3 by 2x+1 and combine like terms.

$$2x^{2}-5x-3-\left(x+3\right)\left(x-1\right)=\left(x-2\right)\times 6$$

Use the distributive property to multiply x+3 by x-1 and combine like terms.

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

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

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

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

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

Combine -5x and -2x to get -7x.

$$x^{2}-7x-3+3=\left(x-2\right)\times 6$$

Add -3 and 3 to get 0.

$$x^{2}-7x=\left(x-2\right)\times 6$$

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

$$x^{2}-7x=6x-12$$

Subtract 6x from both sides.

$$x^{2}-7x-6x=-12$$

Combine -7x and -6x to get -13x.

$$x^{2}-13x=-12$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-12+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-12+\frac{169}{4}$$

Add -12 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{121}{4}$$

Factor x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{11}{2} x-\frac{13}{2}=-\frac{11}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=12 x=1$$