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

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

Solve for x

$$x=-1$$

$$x = \frac{20}{17} = 1\frac{3}{17} \approx 1.176470588$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+2\right), the least common multiple of 2,x+2,3x+6,x^{2}+x-2.

$$6\left(x-1\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6 by x-1.

$$\left(6x-6\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$\left(6x^{2}+6x-12\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x^{2}+6x-12 by \frac{1}{2}.

$$3x^{2}+3x-6+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x-6 by 2x-1 and combine like terms.

$$3x^{2}+3x-6+12x^{2}-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x^{2} and 12x^{2} to get 15x^{2}.

$$15x^{2}+3x-6-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x and -18x to get -15x.

$$15x^{2}-15x-6+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Add -6 and 6 to get 0.

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=12-6x$$

Use the distributive property to multiply 2x-2 by x+4 and combine like terms.

$$15x^{2}-15x+2x^{2}+6x-8=12-6x$$

Combine 15x^{2} and 2x^{2} to get 17x^{2}.

$$17x^{2}-15x+6x-8=12-6x$$

Combine -15x and 6x to get -9x.

$$17x^{2}-9x-8=12-6x$$

Subtract 12 from both sides.

$$17x^{2}-9x-8-12=-6x$$

Subtract 12 from -8 to get -20.

$$17x^{2}-9x-20=-6x$$

Add 6x to both sides.

$$17x^{2}-9x-20+6x=0$$

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

$$17x^{2}-3x-20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 17x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

$$a+b=-3 ab=17\left(-20\right)=-340$$

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. List all such integer pairs that give product -340.

$$1,-340 2,-170 4,-85 5,-68 10,-34 17,-20$$

Calculate the sum for each pair.

$$1-340=-339 2-170=-168 4-85=-81 5-68=-63 10-34=-24 17-20=-3$$

The solution is the pair that gives sum -3.

$$a=-20 b=17$$

Rewrite 17x^{2}-3x-20 as \left(17x^{2}-20x\right)+\left(17x-20\right).

$$\left(17x^{2}-20x\right)+\left(17x-20\right)$$

Factor out x in 17x^{2}-20x.

$$x\left(17x-20\right)+17x-20$$

Factor out common term 17x-20 by using distributive property.

$$\left(17x-20\right)\left(x+1\right)$$

To find equation solutions, solve 17x-20=0 and x+1=0.

$$x=\frac{20}{17} x=-1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+2\right), the least common multiple of 2,x+2,3x+6,x^{2}+x-2.

$$6\left(x-1\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6 by x-1.

$$\left(6x-6\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$\left(6x^{2}+6x-12\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x^{2}+6x-12 by \frac{1}{2}.

$$3x^{2}+3x-6+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x-6 by 2x-1 and combine like terms.

$$3x^{2}+3x-6+12x^{2}-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x^{2} and 12x^{2} to get 15x^{2}.

$$15x^{2}+3x-6-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x and -18x to get -15x.

$$15x^{2}-15x-6+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Add -6 and 6 to get 0.

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=12-6x$$

Use the distributive property to multiply 2x-2 by x+4 and combine like terms.

$$15x^{2}-15x+2x^{2}+6x-8=12-6x$$

Combine 15x^{2} and 2x^{2} to get 17x^{2}.

$$17x^{2}-15x+6x-8=12-6x$$

Combine -15x and 6x to get -9x.

$$17x^{2}-9x-8=12-6x$$

Subtract 12 from both sides.

$$17x^{2}-9x-8-12=-6x$$

Subtract 12 from -8 to get -20.

$$17x^{2}-9x-20=-6x$$

Add 6x to both sides.

$$17x^{2}-9x-20+6x=0$$

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

$$17x^{2}-3x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 17 for a, -3 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 17\left(-20\right)}}{2\times 17}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 17\left(-20\right)}}{2\times 17}$$

Multiply -4 times 17.

$$x=\frac{-\left(-3\right)±\sqrt{9-68\left(-20\right)}}{2\times 17}$$

Multiply -68 times -20.

$$x=\frac{-\left(-3\right)±\sqrt{9+1360}}{2\times 17}$$

Add 9 to 1360.

$$x=\frac{-\left(-3\right)±\sqrt{1369}}{2\times 17}$$

Take the square root of 1369.

$$x=\frac{-\left(-3\right)±37}{2\times 17}$$

The opposite of -3 is 3.

$$x=\frac{3±37}{2\times 17}$$

Multiply 2 times 17.

$$x=\frac{3±37}{34}$$

Now solve the equation x=\frac{3±37}{34} when ± is plus. Add 3 to 37.

$$x=\frac{40}{34}$$

Reduce the fraction \frac{40}{34} to lowest terms by extracting and canceling out 2.

$$x=\frac{20}{17}$$

Now solve the equation x=\frac{3±37}{34} when ± is minus. Subtract 37 from 3.

$$x=-\frac{34}{34}$$

Divide -34 by 34.

$$x=-1$$

The equation is now solved.

$$x=\frac{20}{17} x=-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+2\right), the least common multiple of 2,x+2,3x+6,x^{2}+x-2.

$$6\left(x-1\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6 by x-1.

$$\left(6x-6\right)\left(x+2\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$\left(6x^{2}+6x-12\right)\times \frac{1}{2}+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x^{2}+6x-12 by \frac{1}{2}.

$$3x^{2}+3x-6+\left(6x-6\right)\left(2x-1\right)+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Use the distributive property to multiply 6x-6 by 2x-1 and combine like terms.

$$3x^{2}+3x-6+12x^{2}-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x^{2} and 12x^{2} to get 15x^{2}.

$$15x^{2}+3x-6-18x+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Combine 3x and -18x to get -15x.

$$15x^{2}-15x-6+6+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

Add -6 and 6 to get 0.

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=6\left(2-x\right)$$

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

$$15x^{2}-15x+\left(2x-2\right)\left(x+4\right)=12-6x$$

Use the distributive property to multiply 2x-2 by x+4 and combine like terms.

$$15x^{2}-15x+2x^{2}+6x-8=12-6x$$

Combine 15x^{2} and 2x^{2} to get 17x^{2}.

$$17x^{2}-15x+6x-8=12-6x$$

Combine -15x and 6x to get -9x.

$$17x^{2}-9x-8=12-6x$$

Add 6x to both sides.

$$17x^{2}-9x-8+6x=12$$

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

$$17x^{2}-3x-8=12$$

Add 8 to both sides.

$$17x^{2}-3x=12+8$$

Add 12 and 8 to get 20.

$$17x^{2}-3x=20$$

Divide both sides by 17.

$$\frac{17x^{2}-3x}{17}=\frac{20}{17}$$

Dividing by 17 undoes the multiplication by 17.

$$x^{2}-\frac{3}{17}x=\frac{20}{17}$$

Divide -\frac{3}{17}, the coefficient of the x term, by 2 to get -\frac{3}{34}. Then add the square of -\frac{3}{34} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{17}x+\left(-\frac{3}{34}\right)^{2}=\frac{20}{17}+\left(-\frac{3}{34}\right)^{2}$$

Square -\frac{3}{34} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{17}x+\frac{9}{1156}=\frac{20}{17}+\frac{9}{1156}$$

Add \frac{20}{17} to \frac{9}{1156} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{17}x+\frac{9}{1156}=\frac{1369}{1156}$$

Factor x^{2}-\frac{3}{17}x+\frac{9}{1156}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{34}\right)^{2}=\frac{1369}{1156}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{34}\right)^{2}}=\sqrt{\frac{1369}{1156}}$$

Simplify.

$$x-\frac{3}{34}=\frac{37}{34} x-\frac{3}{34}=-\frac{37}{34}$$

Add \frac{3}{34} to both sides of the equation.

$$x=\frac{20}{17} x=-1$$