Solve 3/x+2-9x/x^2-4=x+1/x-2

$$\frac { 3 } { x + 2 } - \frac { 9 x } { x ^ { 2 } - 4 } = \frac { x + 1 } { x - 2 }$$

Solve for x

$$x=-8$$

$$x=-1$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x^{2}-4,x-2.

$$\left(x-2\right)\times 3-9x=\left(x+2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by 3.

$$3x-6-9x=\left(x+2\right)\left(x+1\right)$$

Combine 3x and -9x to get -6x.

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

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

$$-6x-6=x^{2}+3x+2$$

Subtract x^{2} from both sides.

$$-6x-6-x^{2}=3x+2$$

Subtract 3x from both sides.

$$-6x-6-x^{2}-3x=2$$

Combine -6x and -3x to get -9x.

$$-9x-6-x^{2}=2$$

Subtract 2 from both sides.

$$-9x-6-x^{2}-2=0$$

Subtract 2 from -6 to get -8.

$$-9x-8-x^{2}=0$$

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

$$-x^{2}-9x-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-\left(-8\right)=8$$

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 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -9.

$$a=-1 b=-8$$

Rewrite -x^{2}-9x-8 as \left(-x^{2}-x\right)+\left(-8x-8\right).

$$\left(-x^{2}-x\right)+\left(-8x-8\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x-1\right)+8\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+8\right)$$

To find equation solutions, solve -x-1=0 and x+8=0.

$$x=-1 x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x^{2}-4,x-2.

$$\left(x-2\right)\times 3-9x=\left(x+2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by 3.

$$3x-6-9x=\left(x+2\right)\left(x+1\right)$$

Combine 3x and -9x to get -6x.

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

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

$$-6x-6=x^{2}+3x+2$$

Subtract x^{2} from both sides.

$$-6x-6-x^{2}=3x+2$$

Subtract 3x from both sides.

$$-6x-6-x^{2}-3x=2$$

Combine -6x and -3x to get -9x.

$$-9x-6-x^{2}=2$$

Subtract 2 from both sides.

$$-9x-6-x^{2}-2=0$$

Subtract 2 from -6 to get -8.

$$-9x-8-x^{2}=0$$

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.

$$-x^{2}-9x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -9 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4\left(-8\right)}}{2\left(-1\right)}$$

Multiply 4 times -8.

$$x=\frac{-\left(-9\right)±\sqrt{81-32}}{2\left(-1\right)}$$

Add 81 to -32.

$$x=\frac{-\left(-9\right)±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-\left(-9\right)±7}{2\left(-1\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{9±7}{-2}$$

Now solve the equation x=\frac{9±7}{-2} when ± is plus. Add 9 to 7.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{9±7}{-2} when ± is minus. Subtract 7 from 9.

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

Divide 2 by -2.

$$x=-1$$

The equation is now solved.

$$x=-8 x=-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x+2,x^{2}-4,x-2.

$$\left(x-2\right)\times 3-9x=\left(x+2\right)\left(x+1\right)$$

Use the distributive property to multiply x-2 by 3.

$$3x-6-9x=\left(x+2\right)\left(x+1\right)$$

Combine 3x and -9x to get -6x.

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

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

$$-6x-6=x^{2}+3x+2$$

Subtract x^{2} from both sides.

$$-6x-6-x^{2}=3x+2$$

Subtract 3x from both sides.

$$-6x-6-x^{2}-3x=2$$

Combine -6x and -3x to get -9x.

$$-9x-6-x^{2}=2$$

Add 6 to both sides.

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

Add 2 and 6 to get 8.

$$-9x-x^{2}=8$$

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.

$$-x^{2}-9x=8$$

Divide both sides by -1.

$$\frac{-x^{2}-9x}{-1}=\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{9}{-1}\right)x=\frac{8}{-1}$$

Divide -9 by -1.

$$x^{2}+9x=\frac{8}{-1}$$

Divide 8 by -1.

$$x^{2}+9x=-8$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-8+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=-8+\frac{81}{4}$$

Add -8 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{49}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{7}{2} x+\frac{9}{2}=-\frac{7}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=-1 x=-8$$