Solve 1-x/6=(x-2)x/2+2/3

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

Solve for x

$$x=2$$

$$x=-\frac{1}{3}\approx -0.333333333$$

Steps Using the Quadratic Formula

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

$$6-x=3\left(x-2\right)x+4$$

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

$$6-x=\left(3x-6\right)x+4$$

Use the distributive property to multiply 3x-6 by x.

$$6-x=3x^{2}-6x+4$$

Subtract 3x^{2} from both sides.

$$6-x-3x^{2}=-6x+4$$

Add 6x to both sides.

$$6-x-3x^{2}+6x=4$$

Subtract 4 from both sides.

$$6-x-3x^{2}+6x-4=0$$

Subtract 4 from 6 to get 2.

$$2-x-3x^{2}+6x=0$$

Combine -x and 6x to get 5x.

$$2+5x-3x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-3x^{2}+5x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 5 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-5±\sqrt{25+12\times 2}}{2\left(-3\right)}$$

Multiply 12 times 2.

$$x=\frac{-5±\sqrt{25+24}}{2\left(-3\right)}$$

Add 25 to 24.

$$x=\frac{-5±\sqrt{49}}{2\left(-3\right)}$$

Take the square root of 49.

$$x=\frac{-5±7}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-5±7}{-6}$$

Now solve the equation x=\frac{-5±7}{-6} when ± is plus. Add -5 to 7.

$$x=\frac{2}{-6}$$

Reduce the fraction \frac{2}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-5±7}{-6} when ± is minus. Subtract 7 from -5.

$$x=-\frac{12}{-6}$$

Divide -12 by -6.

$$x=2$$

The equation is now solved.

$$x=-\frac{1}{3} x=2$$

Steps for Completing the Square

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

$$6-x=3\left(x-2\right)x+4$$

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

$$6-x=\left(3x-6\right)x+4$$

Use the distributive property to multiply 3x-6 by x.

$$6-x=3x^{2}-6x+4$$

Subtract 3x^{2} from both sides.

$$6-x-3x^{2}=-6x+4$$

Add 6x to both sides.

$$6-x-3x^{2}+6x=4$$

Subtract 6 from both sides.

$$-x-3x^{2}+6x=4-6$$

Subtract 6 from 4 to get -2.

$$-x-3x^{2}+6x=-2$$

Combine -x and 6x to get 5x.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -3.

$$\frac{-3x^{2}+5x}{-3}=-\frac{2}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{5}{-3}x=-\frac{2}{-3}$$

Divide 5 by -3.

$$x^{2}-\frac{5}{3}x=-\frac{2}{-3}$$

Divide -2 by -3.

$$x^{2}-\frac{5}{3}x=\frac{2}{3}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{7}{6} x-\frac{5}{6}=-\frac{7}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$x=2 x=-\frac{1}{3}$$