Solve 4x/3(x+4)-1/2(x-1)-1/2=0

$$\frac { 4 x } { 3 ( x + 4 ) } - \frac { 1 } { 2 ( x - 1 ) } - \frac { 1 } { 2 } = 0$$

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.

$$\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$\left(8x-8\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-8x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$8x^{2}-8x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -8x and -3x to get -11x.

$$8x^{2}-11x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-11x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12-3x^{2}-9x+12=0$$

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

$$5x^{2}-11x-12-9x+12=0$$

Combine -11x and -9x to get -20x.

$$5x^{2}-20x-12+12=0$$

Add -12 and 12 to get 0.

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

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.

$$\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$\left(8x-8\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-8x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$8x^{2}-8x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -8x and -3x to get -11x.

$$8x^{2}-11x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-11x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12-3x^{2}-9x+12=0$$

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

$$5x^{2}-11x-12-9x+12=0$$

Combine -11x and -9x to get -20x.

$$5x^{2}-20x-12+12=0$$

Add -12 and 12 to get 0.

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

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.

$$\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$\left(8x-8\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-8x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$8x^{2}-8x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -8x and -3x to get -11x.

$$8x^{2}-11x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

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

$$8x^{2}-11x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$8x^{2}-11x-12-3x^{2}-9x+12=0$$

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

$$5x^{2}-11x-12-9x+12=0$$

Combine -11x and -9x to get -20x.

$$5x^{2}-20x-12+12=0$$

Add -12 and 12 to get 0.

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

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.

$$x=4 x=0$$