Solve -3x+9/x^2+x-2=4/x^2+6x+8+4/1-x

$$\frac { - 3 x + 9 } { x ^ { 2 } + x - 2 } = \frac { 4 } { x ^ { 2 } + 6 x + 8 } + \frac { 4 } { 1 - x }$$

Solve for x

$$x=-9$$

$$x=-8$$

Steps Using Factoring

Variable x cannot be equal to any of the values -4,-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x^{2}+x-2,x^{2}+6x+8,1-x.

$$\left(x+4\right)\left(-3x+9\right)=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x+4 by -3x+9 and combine like terms.

$$-3x^{2}-3x+36=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}-3x+36=4x-4-\left(2+x\right)\left(4+x\right)\times 4$$

Multiply -1 and 4 to get -4.

$$-3x^{2}-3x+36=4x-4-4\left(2+x\right)\left(4+x\right)$$

Use the distributive property to multiply -4 by 2+x.

$$-3x^{2}-3x+36=4x-4+\left(-8-4x\right)\left(4+x\right)$$

Use the distributive property to multiply -8-4x by 4+x and combine like terms.

$$-3x^{2}-3x+36=4x-4-32-24x-4x^{2}$$

Subtract 32 from -4 to get -36.

$$-3x^{2}-3x+36=4x-36-24x-4x^{2}$$

Combine 4x and -24x to get -20x.

$$-3x^{2}-3x+36=-20x-36-4x^{2}$$

Add 20x to both sides.

$$-3x^{2}-3x+36+20x=-36-4x^{2}$$

Combine -3x and 20x to get 17x.

$$-3x^{2}+17x+36=-36-4x^{2}$$

Subtract -36 from both sides.

$$-3x^{2}+17x+36-\left(-36\right)=-4x^{2}$$

The opposite of -36 is 36.

$$-3x^{2}+17x+36+36=-4x^{2}$$

Combine 36 and 36 to get 2\times 36.

$$-3x^{2}+17x+2\times 36=-4x^{2}$$

Add 4x^{2} to both sides.

$$-3x^{2}+17x+2\times 36+4x^{2}=0$$

Multiply 2 and 36 to get 72.

$$-3x^{2}+17x+72+4x^{2}=0$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}+17x+72=0$$

To solve the equation, factor x^{2}+17x+72 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.

$$a+b=17 ab=72$$

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 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 17.

$$a=8 b=9$$

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

$$\left(x+8\right)\left(x+9\right)$$

To find equation solutions, solve x+8=0 and x+9=0.

$$x=-8 x=-9$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x^{2}+x-2,x^{2}+6x+8,1-x.

$$\left(x+4\right)\left(-3x+9\right)=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x+4 by -3x+9 and combine like terms.

$$-3x^{2}-3x+36=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}-3x+36=4x-4-\left(2+x\right)\left(4+x\right)\times 4$$

Multiply -1 and 4 to get -4.

$$-3x^{2}-3x+36=4x-4-4\left(2+x\right)\left(4+x\right)$$

Use the distributive property to multiply -4 by 2+x.

$$-3x^{2}-3x+36=4x-4+\left(-8-4x\right)\left(4+x\right)$$

Use the distributive property to multiply -8-4x by 4+x and combine like terms.

$$-3x^{2}-3x+36=4x-4-32-24x-4x^{2}$$

Subtract 32 from -4 to get -36.

$$-3x^{2}-3x+36=4x-36-24x-4x^{2}$$

Combine 4x and -24x to get -20x.

$$-3x^{2}-3x+36=-20x-36-4x^{2}$$

Add 20x to both sides.

$$-3x^{2}-3x+36+20x=-36-4x^{2}$$

Combine -3x and 20x to get 17x.

$$-3x^{2}+17x+36=-36-4x^{2}$$

Subtract -36 from both sides.

$$-3x^{2}+17x+36-\left(-36\right)=-4x^{2}$$

The opposite of -36 is 36.

$$-3x^{2}+17x+36+36=-4x^{2}$$

Combine 36 and 36 to get 2\times 36.

$$-3x^{2}+17x+2\times 36=-4x^{2}$$

Add 4x^{2} to both sides.

$$-3x^{2}+17x+2\times 36+4x^{2}=0$$

Multiply 2 and 36 to get 72.

$$-3x^{2}+17x+72+4x^{2}=0$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}+17x+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=17 ab=1\times 72=72$$

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 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 17.

$$a=8 b=9$$

Rewrite x^{2}+17x+72 as \left(x^{2}+8x\right)+\left(9x+72\right).

$$\left(x^{2}+8x\right)+\left(9x+72\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x+8\right)+9\left(x+8\right)$$

Factor out common term x+8 by using distributive property.

$$\left(x+8\right)\left(x+9\right)$$

To find equation solutions, solve x+8=0 and x+9=0.

$$x=-8 x=-9$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x^{2}+x-2,x^{2}+6x+8,1-x.

$$\left(x+4\right)\left(-3x+9\right)=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x+4 by -3x+9 and combine like terms.

$$-3x^{2}-3x+36=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}-3x+36=4x-4-\left(2+x\right)\left(4+x\right)\times 4$$

Multiply -1 and 4 to get -4.

$$-3x^{2}-3x+36=4x-4-4\left(2+x\right)\left(4+x\right)$$

Use the distributive property to multiply -4 by 2+x.

$$-3x^{2}-3x+36=4x-4+\left(-8-4x\right)\left(4+x\right)$$

Use the distributive property to multiply -8-4x by 4+x and combine like terms.

$$-3x^{2}-3x+36=4x-4-32-24x-4x^{2}$$

Subtract 32 from -4 to get -36.

$$-3x^{2}-3x+36=4x-36-24x-4x^{2}$$

Combine 4x and -24x to get -20x.

$$-3x^{2}-3x+36=-20x-36-4x^{2}$$

Add 20x to both sides.

$$-3x^{2}-3x+36+20x=-36-4x^{2}$$

Combine -3x and 20x to get 17x.

$$-3x^{2}+17x+36=-36-4x^{2}$$

Subtract -36 from both sides.

$$-3x^{2}+17x+36-\left(-36\right)=-4x^{2}$$

The opposite of -36 is 36.

$$-3x^{2}+17x+36+36=-4x^{2}$$

Combine 36 and 36 to get 2\times 36.

$$-3x^{2}+17x+2\times 36=-4x^{2}$$

Add 4x^{2} to both sides.

$$-3x^{2}+17x+2\times 36+4x^{2}=0$$

Multiply 2 and 36 to get 72.

$$-3x^{2}+17x+72+4x^{2}=0$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}+17x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 17 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 72}}{2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-17±\sqrt{289-288}}{2}$$

Add 289 to -288.

$$x=\frac{-17±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-17±1}{2}$$

Now solve the equation x=\frac{-17±1}{2} when ± is plus. Add -17 to 1.

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

Divide -16 by 2.

$$x=-8$$

Now solve the equation x=\frac{-17±1}{2} when ± is minus. Subtract 1 from -17.

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

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=-8 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x^{2}+x-2,x^{2}+6x+8,1-x.

$$\left(x+4\right)\left(-3x+9\right)=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x+4 by -3x+9 and combine like terms.

$$-3x^{2}-3x+36=\left(x-1\right)\times 4-\left(2+x\right)\left(4+x\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}-3x+36=4x-4-\left(2+x\right)\left(4+x\right)\times 4$$

Multiply -1 and 4 to get -4.

$$-3x^{2}-3x+36=4x-4-4\left(2+x\right)\left(4+x\right)$$

Use the distributive property to multiply -4 by 2+x.

$$-3x^{2}-3x+36=4x-4+\left(-8-4x\right)\left(4+x\right)$$

Use the distributive property to multiply -8-4x by 4+x and combine like terms.

$$-3x^{2}-3x+36=4x-4-32-24x-4x^{2}$$

Subtract 32 from -4 to get -36.

$$-3x^{2}-3x+36=4x-36-24x-4x^{2}$$

Combine 4x and -24x to get -20x.

$$-3x^{2}-3x+36=-20x-36-4x^{2}$$

Add 20x to both sides.

$$-3x^{2}-3x+36+20x=-36-4x^{2}$$

Combine -3x and 20x to get 17x.

$$-3x^{2}+17x+36=-36-4x^{2}$$

Add 4x^{2} to both sides.

$$-3x^{2}+17x+36+4x^{2}=-36$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}+17x+36=-36$$

Subtract 36 from both sides.

$$x^{2}+17x=-36-36$$

Subtract 36 from -36 to get -72.

$$x^{2}+17x=-72$$

Divide 17, the coefficient of the x term, by 2 to get \frac{17}{2}. Then add the square of \frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+17x+\left(\frac{17}{2}\right)^{2}=-72+\left(\frac{17}{2}\right)^{2}$$

Square \frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+17x+\frac{289}{4}=-72+\frac{289}{4}$$

Add -72 to \frac{289}{4}.

$$x^{2}+17x+\frac{289}{4}=\frac{1}{4}$$

Factor x^{2}+17x+\frac{289}{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}.

$$\left(x+\frac{17}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x+\frac{17}{2}=\frac{1}{2} x+\frac{17}{2}=-\frac{1}{2}$$

Subtract \frac{17}{2} from both sides of the equation.

$$x=-8 x=-9$$