Solve 4x/3(x+4)=1/2(x-1)+1/2

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

Solve for x

$$x=4$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$\left(2x-2\right)\times 4x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

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

$$\left(8x-8\right)x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Use the distributive property to multiply 8x-8 by x.

$$8x^{2}-8x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Multiply 6 and \frac{1}{2} to get 3.

$$8x^{2}-8x=3x+12+3\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply 3 by x-1.

$$8x^{2}-8x=3x+12+\left(3x-3\right)\left(x+4\right)$$

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

$$8x^{2}-8x=3x+12+3x^{2}+9x-12$$

Combine 3x and 9x to get 12x.

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

Subtract 12 from 12 to get 0.

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

Subtract 12x from both sides.

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

Combine -8x and -12x to get -20x.

$$8x^{2}-20x=3x^{2}$$

Subtract 3x^{2} from both sides.

$$8x^{2}-20x-3x^{2}=0$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}-20x=0$$

Factor out x.

$$x\left(5x-20\right)=0$$

To find equation solutions, solve x=0 and 5x-20=0.

$$x=0 x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$\left(2x-2\right)\times 4x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

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

$$\left(8x-8\right)x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Use the distributive property to multiply 8x-8 by x.

$$8x^{2}-8x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Multiply 6 and \frac{1}{2} to get 3.

$$8x^{2}-8x=3x+12+3\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply 3 by x-1.

$$8x^{2}-8x=3x+12+\left(3x-3\right)\left(x+4\right)$$

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

$$8x^{2}-8x=3x+12+3x^{2}+9x-12$$

Combine 3x and 9x to get 12x.

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

Subtract 12 from 12 to get 0.

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

Subtract 12x from both sides.

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

Combine -8x and -12x to get -20x.

$$8x^{2}-20x=3x^{2}$$

Subtract 3x^{2} from both sides.

$$8x^{2}-20x-3x^{2}=0$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}-20x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -20 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}}}{2\times 5}$$

Take the square root of \left(-20\right)^{2}.

$$x=\frac{-\left(-20\right)±20}{2\times 5}$$

The opposite of -20 is 20.

$$x=\frac{20±20}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{20±20}{10}$$

Now solve the equation x=\frac{20±20}{10} when ± is plus. Add 20 to 20.

$$x=\frac{40}{10}$$

Divide 40 by 10.

$$x=4$$

Now solve the equation x=\frac{20±20}{10} when ± is minus. Subtract 20 from 20.

$$x=\frac{0}{10}$$

Divide 0 by 10.

$$x=0$$

The equation is now solved.

$$x=4 x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$\left(2x-2\right)\times 4x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

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

$$\left(8x-8\right)x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Use the distributive property to multiply 8x-8 by x.

$$8x^{2}-8x=3x+12+6\left(x-1\right)\left(x+4\right)\times \frac{1}{2}$$

Multiply 6 and \frac{1}{2} to get 3.

$$8x^{2}-8x=3x+12+3\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply 3 by x-1.

$$8x^{2}-8x=3x+12+\left(3x-3\right)\left(x+4\right)$$

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

$$8x^{2}-8x=3x+12+3x^{2}+9x-12$$

Combine 3x and 9x to get 12x.

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

Subtract 12 from 12 to get 0.

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

Subtract 12x from both sides.

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

Combine -8x and -12x to get -20x.

$$8x^{2}-20x=3x^{2}$$

Subtract 3x^{2} from both sides.

$$8x^{2}-20x-3x^{2}=0$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}-20x=0$$

Divide both sides by 5.

$$\frac{5x^{2}-20x}{5}=\frac{0}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{20}{5}\right)x=\frac{0}{5}$$

Divide -20 by 5.

$$x^{2}-4x=\frac{0}{5}$$

Divide 0 by 5.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=2 x-2=-2$$

Add 2 to both sides of the equation.

$$x=4 x=0$$