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

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

Solve for x

$$x=\frac{1}{12}\approx 0.083333333$$

Steps Using Factoring

Variable x cannot be equal to any of the values -2,0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x\left(2x-1\right)\left(x+2\right), the least common multiple of 2x+4,1-2x,x,x^{2}+2x.

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

Use the distributive property to multiply x by 2x-1.

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

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

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

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

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

Combine -3x and 4x to get x.

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

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$8x^{2}+x+2\left(2x-1\right)\left(x+2\right)=\left(4x-2\right)\times 2$$

Use the distributive property to multiply 2 by 2x-1.

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

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

$$8x^{2}+x+4x^{2}+6x-4=\left(4x-2\right)\times 2$$

Combine 8x^{2} and 4x^{2} to get 12x^{2}.

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

Combine x and 6x to get 7x.

$$12x^{2}+7x-4=\left(4x-2\right)\times 2$$

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

$$12x^{2}+7x-4=8x-4$$

Subtract 8x from both sides.

$$12x^{2}+7x-4-8x=-4$$

Combine 7x and -8x to get -x.

$$12x^{2}-x-4=-4$$

Add 4 to both sides.

$$12x^{2}-x-4+4=0$$

Add -4 and 4 to get 0.

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

Factor out x.

$$x\left(12x-1\right)=0$$

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

$$x=0 x=\frac{1}{12}$$

Variable x cannot be equal to 0.

$$x=\frac{1}{12}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x\left(2x-1\right)\left(x+2\right), the least common multiple of 2x+4,1-2x,x,x^{2}+2x.

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

Use the distributive property to multiply x by 2x-1.

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

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

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

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

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

Combine -3x and 4x to get x.

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

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$8x^{2}+x+2\left(2x-1\right)\left(x+2\right)=\left(4x-2\right)\times 2$$

Use the distributive property to multiply 2 by 2x-1.

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

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

$$8x^{2}+x+4x^{2}+6x-4=\left(4x-2\right)\times 2$$

Combine 8x^{2} and 4x^{2} to get 12x^{2}.

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

Combine x and 6x to get 7x.

$$12x^{2}+7x-4=\left(4x-2\right)\times 2$$

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

$$12x^{2}+7x-4=8x-4$$

Subtract 8x from both sides.

$$12x^{2}+7x-4-8x=-4$$

Combine 7x and -8x to get -x.

$$12x^{2}-x-4=-4$$

Add 4 to both sides.

$$12x^{2}-x-4+4=0$$

Add -4 and 4 to get 0.

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

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

$$x=\frac{-\left(-1\right)±\sqrt{1}}{2\times 12}$$

Take the square root of 1.

$$x=\frac{-\left(-1\right)±1}{2\times 12}$$

The opposite of -1 is 1.

$$x=\frac{1±1}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{1±1}{24}$$

Now solve the equation x=\frac{1±1}{24} when ± is plus. Add 1 to 1.

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

Reduce the fraction \frac{2}{24} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{12}$$

Now solve the equation x=\frac{1±1}{24} when ± is minus. Subtract 1 from 1.

$$x=\frac{0}{24}$$

Divide 0 by 24.

$$x=0$$

The equation is now solved.

$$x=\frac{1}{12} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{1}{12}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x\left(2x-1\right)\left(x+2\right), the least common multiple of 2x+4,1-2x,x,x^{2}+2x.

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

Use the distributive property to multiply x by 2x-1.

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

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

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

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

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

Combine -3x and 4x to get x.

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

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$8x^{2}+x+2\left(2x-1\right)\left(x+2\right)=\left(4x-2\right)\times 2$$

Use the distributive property to multiply 2 by 2x-1.

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

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

$$8x^{2}+x+4x^{2}+6x-4=\left(4x-2\right)\times 2$$

Combine 8x^{2} and 4x^{2} to get 12x^{2}.

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

Combine x and 6x to get 7x.

$$12x^{2}+7x-4=\left(4x-2\right)\times 2$$

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

$$12x^{2}+7x-4=8x-4$$

Subtract 8x from both sides.

$$12x^{2}+7x-4-8x=-4$$

Combine 7x and -8x to get -x.

$$12x^{2}-x-4=-4$$

Add 4 to both sides.

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

Add -4 and 4 to get 0.

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

Divide both sides by 12.

$$\frac{12x^{2}-x}{12}=\frac{0}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}-\frac{1}{12}x=\frac{0}{12}$$

Divide 0 by 12.

$$x^{2}-\frac{1}{12}x=0$$

Divide -\frac{1}{12}, the coefficient of the x term, by 2 to get -\frac{1}{24}. Then add the square of -\frac{1}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{12}x+\left(-\frac{1}{24}\right)^{2}=\left(-\frac{1}{24}\right)^{2}$$

Square -\frac{1}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{12}x+\frac{1}{576}=\frac{1}{576}$$

Factor x^{2}-\frac{1}{12}x+\frac{1}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{24}\right)^{2}=\frac{1}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{24}\right)^{2}}=\sqrt{\frac{1}{576}}$$

Simplify.

$$x-\frac{1}{24}=\frac{1}{24} x-\frac{1}{24}=-\frac{1}{24}$$

Add \frac{1}{24} to both sides of the equation.

$$x=\frac{1}{12} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{1}{12}$$