Solve {c}{x=+x/x=x=1}{x-2x(x+2)/3+7/6x-2x(x-1)/3=frac{4}{3}-frac{2(x-1)^2}{6}}

$$\left. \begin{array} { c } { x = + \frac { x } { x } = x = 1 } \\ { \frac { x - 2 x ( x + 2 ) } { 3 } + \frac { 7 } { 6 } x - \frac { 2 x ( x - 1 ) } { 3 } = \frac { 4 } { 3 } - \frac { 2 ( x - 1 ) ^ { 2 } } { 6 } } \end{array} \right.$$

Solve for x

$$x=1$$

Steps Using Factoring

Subtract \frac{x}{x} from both sides.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{xx}{x}-\frac{x}{x}=0$$

Since \frac{xx}{x} and \frac{x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{xx-x}{x}=0$$

Do the multiplications in xx-x.

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

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

Factor out x.

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

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

$$x=0 x=1$$

Variable x cannot be equal to 0.

$$x=1$$

Steps Using the Quadratic Formula

Subtract \frac{x}{x} from both sides.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{xx}{x}-\frac{x}{x}=0$$

Since \frac{xx}{x} and \frac{x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{xx-x}{x}=0$$

Do the multiplications in xx-x.

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

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

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

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

Take the square root of 1.

$$x=\frac{-\left(-1\right)±1}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±1}{2}$$

Now solve the equation x=\frac{1±1}{2} when ± is plus. Add 1 to 1.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{1±1}{2} when ± is minus. Subtract 1 from 1.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

The equation is now solved.

$$x=1 x=0$$

Variable x cannot be equal to 0.

$$x=1$$

Steps for Completing the Square

Subtract \frac{x}{x} from both sides.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{xx}{x}-\frac{x}{x}=0$$

Since \frac{xx}{x} and \frac{x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{xx-x}{x}=0$$

Do the multiplications in xx-x.

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

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Factor x^{2}-x+\frac{1}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{1}{2}=\frac{1}{2} x-\frac{1}{2}=-\frac{1}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=1 x=0$$

Variable x cannot be equal to 0.

$$x=1$$