Solve x/2=2/3+7/6x

$$\frac { x } { 2 } = \frac { 2 } { 3 } + \frac { 7 } { 6 x }$$

Solve for x

$$x=-1$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 2,3,6x.

$$3xx=6x\times \frac{2}{3}+7$$

Multiply x and x to get x^{2}.

$$3x^{2}=6x\times \frac{2}{3}+7$$

Multiply 6 and \frac{2}{3} to get 4.

$$3x^{2}=4x+7$$

Subtract 4x from both sides.

$$3x^{2}-4x=7$$

Subtract 7 from both sides.

$$3x^{2}-4x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-7\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-7\right)}}{2\times 3}$$

Multiply -12 times -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+84}}{2\times 3}$$

Add 16 to 84.

$$x=\frac{-\left(-4\right)±\sqrt{100}}{2\times 3}$$

Take the square root of 100.

$$x=\frac{-\left(-4\right)±10}{2\times 3}$$

The opposite of -4 is 4.

$$x=\frac{4±10}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{4±10}{6}$$

Now solve the equation x=\frac{4±10}{6} when ± is plus. Add 4 to 10.

$$x=\frac{14}{6}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{4±10}{6} when ± is minus. Subtract 10 from 4.

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

Divide -6 by 6.

$$x=-1$$

The equation is now solved.

$$x=\frac{7}{3} x=-1$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 2,3,6x.

$$3xx=6x\times \frac{2}{3}+7$$

Multiply x and x to get x^{2}.

$$3x^{2}=6x\times \frac{2}{3}+7$$

Multiply 6 and \frac{2}{3} to get 4.

$$3x^{2}=4x+7$$

Subtract 4x from both sides.

$$3x^{2}-4x=7$$

Divide both sides by 3.

$$\frac{3x^{2}-4x}{3}=\frac{7}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{4}{3}x=\frac{7}{3}$$

Divide -\frac{4}{3}, the coefficient of the x term, by 2 to get -\frac{2}{3}. Then add the square of -\frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=\frac{7}{3}+\left(-\frac{2}{3}\right)^{2}$$

Square -\frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{7}{3}+\frac{4}{9}$$

Add \frac{7}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{25}{9}$$

Factor x^{2}-\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{25}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Simplify.

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

Add \frac{2}{3} to both sides of the equation.

$$x=\frac{7}{3} x=-1$$