Solve 1/6x/2x+5=x/3+2x

$$\frac{ \frac{ 1 }{ 6 } x }{ 2x+5 } = \frac{ x }{ 3+2x }$$

Solve for x

$$x = -\frac{27}{10} = -2\frac{7}{10} = -2.7$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -\frac{5}{2},-\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x+3\right)\left(2x+5\right), the least common multiple of 2x+5,3+2x.

$$\left(2x+3\right)\times \frac{1}{6}x=\left(2x+5\right)x$$

Use the distributive property to multiply 2x+3 by \frac{1}{6}.

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

Use the distributive property to multiply \frac{1}{3}x+\frac{1}{2} by x.

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

Use the distributive property to multiply 2x+5 by x.

$$\frac{1}{3}x^{2}+\frac{1}{2}x=2x^{2}+5x$$

Subtract 2x^{2} from both sides.

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

Combine \frac{1}{3}x^{2} and -2x^{2} to get -\frac{5}{3}x^{2}.

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

Subtract 5x from both sides.

$$-\frac{5}{3}x^{2}+\frac{1}{2}x-5x=0$$

Combine \frac{1}{2}x and -5x to get -\frac{9}{2}x.

$$-\frac{5}{3}x^{2}-\frac{9}{2}x=0$$

Factor out x.

$$x\left(-\frac{5}{3}x-\frac{9}{2}\right)=0$$

To find equation solutions, solve x=0 and -\frac{5x}{3}-\frac{9}{2}=0.

$$x=0 x=-\frac{27}{10}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{5}{2},-\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x+3\right)\left(2x+5\right), the least common multiple of 2x+5,3+2x.

$$\left(2x+3\right)\times \frac{1}{6}x=\left(2x+5\right)x$$

Use the distributive property to multiply 2x+3 by \frac{1}{6}.

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

Use the distributive property to multiply \frac{1}{3}x+\frac{1}{2} by x.

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

Use the distributive property to multiply 2x+5 by x.

$$\frac{1}{3}x^{2}+\frac{1}{2}x=2x^{2}+5x$$

Subtract 2x^{2} from both sides.

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

Combine \frac{1}{3}x^{2} and -2x^{2} to get -\frac{5}{3}x^{2}.

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

Subtract 5x from both sides.

$$-\frac{5}{3}x^{2}+\frac{1}{2}x-5x=0$$

Combine \frac{1}{2}x and -5x to get -\frac{9}{2}x.

$$-\frac{5}{3}x^{2}-\frac{9}{2}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{5}{3} for a, -\frac{9}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\left(-\frac{9}{2}\right)^{2}}}{2\left(-\frac{5}{3}\right)}$$

Take the square root of \left(-\frac{9}{2}\right)^{2}.

$$x=\frac{-\left(-\frac{9}{2}\right)±\frac{9}{2}}{2\left(-\frac{5}{3}\right)}$$

The opposite of -\frac{9}{2} is \frac{9}{2}.

$$x=\frac{\frac{9}{2}±\frac{9}{2}}{2\left(-\frac{5}{3}\right)}$$

Multiply 2 times -\frac{5}{3}.

$$x=\frac{\frac{9}{2}±\frac{9}{2}}{-\frac{10}{3}}$$

Now solve the equation x=\frac{\frac{9}{2}±\frac{9}{2}}{-\frac{10}{3}} when ± is plus. Add \frac{9}{2} to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{9}{-\frac{10}{3}}$$

Divide 9 by -\frac{10}{3} by multiplying 9 by the reciprocal of -\frac{10}{3}.

$$x=-\frac{27}{10}$$

Now solve the equation x=\frac{\frac{9}{2}±\frac{9}{2}}{-\frac{10}{3}} when ± is minus. Subtract \frac{9}{2} from \frac{9}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-\frac{10}{3}}$$

Divide 0 by -\frac{10}{3} by multiplying 0 by the reciprocal of -\frac{10}{3}.

$$x=0$$

The equation is now solved.

$$x=-\frac{27}{10} x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{5}{2},-\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x+3\right)\left(2x+5\right), the least common multiple of 2x+5,3+2x.

$$\left(2x+3\right)\times \frac{1}{6}x=\left(2x+5\right)x$$

Use the distributive property to multiply 2x+3 by \frac{1}{6}.

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

Use the distributive property to multiply \frac{1}{3}x+\frac{1}{2} by x.

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

Use the distributive property to multiply 2x+5 by x.

$$\frac{1}{3}x^{2}+\frac{1}{2}x=2x^{2}+5x$$

Subtract 2x^{2} from both sides.

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

Combine \frac{1}{3}x^{2} and -2x^{2} to get -\frac{5}{3}x^{2}.

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

Subtract 5x from both sides.

$$-\frac{5}{3}x^{2}+\frac{1}{2}x-5x=0$$

Combine \frac{1}{2}x and -5x to get -\frac{9}{2}x.

$$-\frac{5}{3}x^{2}-\frac{9}{2}x=0$$

Divide both sides of the equation by -\frac{5}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{5}{3}x^{2}-\frac{9}{2}x}{-\frac{5}{3}}=\frac{0}{-\frac{5}{3}}$$

Dividing by -\frac{5}{3} undoes the multiplication by -\frac{5}{3}.

$$x^{2}+\left(-\frac{\frac{9}{2}}{-\frac{5}{3}}\right)x=\frac{0}{-\frac{5}{3}}$$

Divide -\frac{9}{2} by -\frac{5}{3} by multiplying -\frac{9}{2} by the reciprocal of -\frac{5}{3}.

$$x^{2}+\frac{27}{10}x=\frac{0}{-\frac{5}{3}}$$

Divide 0 by -\frac{5}{3} by multiplying 0 by the reciprocal of -\frac{5}{3}.

$$x^{2}+\frac{27}{10}x=0$$

Divide \frac{27}{10}, the coefficient of the x term, by 2 to get \frac{27}{20}. Then add the square of \frac{27}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{27}{10}x+\left(\frac{27}{20}\right)^{2}=\left(\frac{27}{20}\right)^{2}$$

Square \frac{27}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{27}{10}x+\frac{729}{400}=\frac{729}{400}$$

Factor x^{2}+\frac{27}{10}x+\frac{729}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{27}{20}\right)^{2}=\frac{729}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{20}\right)^{2}}=\sqrt{\frac{729}{400}}$$

Simplify.

$$x+\frac{27}{20}=\frac{27}{20} x+\frac{27}{20}=-\frac{27}{20}$$

Subtract \frac{27}{20} from both sides of the equation.

$$x=0 x=-\frac{27}{10}$$