Solve (x+2/3)(x-3/2)-x^2-4/6=1/3+frac{2x-1}{6}-frac{1}{2}(x+1)

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

Solve for x

$$x=\frac{4}{5}=0.8$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 6, the least common multiple of 3,2,6.

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

Use the distributive property to multiply 6 by x+\frac{2}{3}.

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

Use the distributive property to multiply 6x+4 by x-\frac{3}{2} and combine like terms.

$$6x^{2}-5x-6-\left(x^{2}-4\right)=2+2x-1-3\left(x+1\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$6x^{2}-5x-6-x^{2}+4=2+2x-1-3\left(x+1\right)$$

Combine 6x^{2} and -x^{2} to get 5x^{2}.

$$5x^{2}-5x-6+4=2+2x-1-3\left(x+1\right)$$

Add -6 and 4 to get -2.

$$5x^{2}-5x-2=2+2x-1-3\left(x+1\right)$$

Subtract 1 from 2 to get 1.

$$5x^{2}-5x-2=1+2x-3\left(x+1\right)$$

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

$$5x^{2}-5x-2=1+2x-3x-3$$

Combine 2x and -3x to get -x.

$$5x^{2}-5x-2=1-x-3$$

Subtract 3 from 1 to get -2.

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

Subtract -2 from both sides.

$$5x^{2}-5x-2-\left(-2\right)=-x$$

The opposite of -2 is 2.

$$5x^{2}-5x-2+2=-x$$

Add x to both sides.

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

Add -2 and 2 to get 0.

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

Combine -5x and x to get -4x.

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

Factor out x.

$$x\left(5x-4\right)=0$$

To find equation solutions, solve x=0 and 5x-4=0.

$$x=0 x=\frac{4}{5}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2,6.

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

Use the distributive property to multiply 6 by x+\frac{2}{3}.

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

Use the distributive property to multiply 6x+4 by x-\frac{3}{2} and combine like terms.

$$6x^{2}-5x-6-\left(x^{2}-4\right)=2+2x-1-3\left(x+1\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$6x^{2}-5x-6-x^{2}+4=2+2x-1-3\left(x+1\right)$$

Combine 6x^{2} and -x^{2} to get 5x^{2}.

$$5x^{2}-5x-6+4=2+2x-1-3\left(x+1\right)$$

Add -6 and 4 to get -2.

$$5x^{2}-5x-2=2+2x-1-3\left(x+1\right)$$

Subtract 1 from 2 to get 1.

$$5x^{2}-5x-2=1+2x-3\left(x+1\right)$$

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

$$5x^{2}-5x-2=1+2x-3x-3$$

Combine 2x and -3x to get -x.

$$5x^{2}-5x-2=1-x-3$$

Subtract 3 from 1 to get -2.

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

Subtract -2 from both sides.

$$5x^{2}-5x-2-\left(-2\right)=-x$$

The opposite of -2 is 2.

$$5x^{2}-5x-2+2=-x$$

Add x to both sides.

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

Add -2 and 2 to get 0.

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

Combine -5x and x to get -4x.

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

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

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2\times 5}$$

Take the square root of \left(-4\right)^{2}.

$$x=\frac{-\left(-4\right)±4}{2\times 5}$$

The opposite of -4 is 4.

$$x=\frac{4±4}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{4±4}{10}$$

Now solve the equation x=\frac{4±4}{10} when ± is plus. Add 4 to 4.

$$x=\frac{8}{10}$$

Reduce the fraction \frac{8}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{5}$$

Now solve the equation x=\frac{4±4}{10} when ± is minus. Subtract 4 from 4.

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

Divide 0 by 10.

$$x=0$$

The equation is now solved.

$$x=\frac{4}{5} x=0$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2,6.

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

Use the distributive property to multiply 6 by x+\frac{2}{3}.

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

Use the distributive property to multiply 6x+4 by x-\frac{3}{2} and combine like terms.

$$6x^{2}-5x-6-\left(x^{2}-4\right)=2+2x-1-3\left(x+1\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$6x^{2}-5x-6-x^{2}+4=2+2x-1-3\left(x+1\right)$$

Combine 6x^{2} and -x^{2} to get 5x^{2}.

$$5x^{2}-5x-6+4=2+2x-1-3\left(x+1\right)$$

Add -6 and 4 to get -2.

$$5x^{2}-5x-2=2+2x-1-3\left(x+1\right)$$

Subtract 1 from 2 to get 1.

$$5x^{2}-5x-2=1+2x-3\left(x+1\right)$$

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

$$5x^{2}-5x-2=1+2x-3x-3$$

Combine 2x and -3x to get -x.

$$5x^{2}-5x-2=1-x-3$$

Subtract 3 from 1 to get -2.

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

Add x to both sides.

$$5x^{2}-5x-2+x=-2$$

Combine -5x and x to get -4x.

$$5x^{2}-4x-2=-2$$

Add 2 to both sides.

$$5x^{2}-4x=-2+2$$

Add -2 and 2 to get 0.

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

Divide both sides by 5.

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

Dividing by 5 undoes the multiplication by 5.

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

Divide 0 by 5.

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

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{5}x+\left(-\frac{2}{5}\right)^{2}=\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=\frac{4}{25}$$

Factor x^{2}-\frac{4}{5}x+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{5}\right)^{2}=\frac{4}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{5}\right)^{2}}=\sqrt{\frac{4}{25}}$$

Simplify.

$$x-\frac{2}{5}=\frac{2}{5} x-\frac{2}{5}=-\frac{2}{5}$$

Add \frac{2}{5} to both sides of the equation.

$$x=\frac{4}{5} x=0$$