Solve y=2y+3/3y-2

$$y= \frac{ 2y+3 }{ 3y-2 }$$

Solve for y

$$y = \frac{\sqrt{13} + 2}{3} \approx 1.868517092$$

$$y=\frac{2-\sqrt{13}}{3}\approx -0.535183758$$

Steps Using the Quadratic Formula

Subtract \frac{2y+3}{3y-2} from both sides.

$$y-\frac{2y+3}{3y-2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{3y-2}{3y-2}.

$$\frac{y\left(3y-2\right)}{3y-2}-\frac{2y+3}{3y-2}=0$$

Since \frac{y\left(3y-2\right)}{3y-2} and \frac{2y+3}{3y-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\left(3y-2\right)-\left(2y+3\right)}{3y-2}=0$$

Do the multiplications in y\left(3y-2\right)-\left(2y+3\right).

$$\frac{3y^{2}-2y-2y-3}{3y-2}=0$$

Combine like terms in 3y^{2}-2y-2y-3.

$$\frac{3y^{2}-4y-3}{3y-2}=0$$

Variable y cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3y-2.

$$3y^{2}-4y-3=0$$

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

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

Square -4.

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

Multiply -4 times 3.

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

Multiply -12 times -3.

$$y=\frac{-\left(-4\right)±\sqrt{16+36}}{2\times 3}$$

Add 16 to 36.

$$y=\frac{-\left(-4\right)±\sqrt{52}}{2\times 3}$$

Take the square root of 52.

$$y=\frac{-\left(-4\right)±2\sqrt{13}}{2\times 3}$$

The opposite of -4 is 4.

$$y=\frac{4±2\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{4±2\sqrt{13}}{6}$$

Now solve the equation y=\frac{4±2\sqrt{13}}{6} when ± is plus. Add 4 to 2\sqrt{13}.

$$y=\frac{2\sqrt{13}+4}{6}$$

Divide 4+2\sqrt{13} by 6.

$$y=\frac{\sqrt{13}+2}{3}$$

Now solve the equation y=\frac{4±2\sqrt{13}}{6} when ± is minus. Subtract 2\sqrt{13} from 4.

$$y=\frac{4-2\sqrt{13}}{6}$$

Divide 4-2\sqrt{13} by 6.

$$y=\frac{2-\sqrt{13}}{3}$$

The equation is now solved.

$$y=\frac{\sqrt{13}+2}{3} y=\frac{2-\sqrt{13}}{3}$$

Steps for Completing the Square

Subtract \frac{2y+3}{3y-2} from both sides.

$$y-\frac{2y+3}{3y-2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{3y-2}{3y-2}.

$$\frac{y\left(3y-2\right)}{3y-2}-\frac{2y+3}{3y-2}=0$$

Since \frac{y\left(3y-2\right)}{3y-2} and \frac{2y+3}{3y-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\left(3y-2\right)-\left(2y+3\right)}{3y-2}=0$$

Do the multiplications in y\left(3y-2\right)-\left(2y+3\right).

$$\frac{3y^{2}-2y-2y-3}{3y-2}=0$$

Combine like terms in 3y^{2}-2y-2y-3.

$$\frac{3y^{2}-4y-3}{3y-2}=0$$

Variable y cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3y-2.

$$3y^{2}-4y-3=0$$

Add 3 to both sides. Anything plus zero gives itself.

$$3y^{2}-4y=3$$

Divide both sides by 3.

$$\frac{3y^{2}-4y}{3}=\frac{3}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}-\frac{4}{3}y=\frac{3}{3}$$

Divide 3 by 3.

$$y^{2}-\frac{4}{3}y=1$$

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.

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

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

$$y^{2}-\frac{4}{3}y+\frac{4}{9}=1+\frac{4}{9}$$

Add 1 to \frac{4}{9}.

$$y^{2}-\frac{4}{3}y+\frac{4}{9}=\frac{13}{9}$$

Factor y^{2}-\frac{4}{3}y+\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}.

$$\left(y-\frac{2}{3}\right)^{2}=\frac{13}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{2}{3}\right)^{2}}=\sqrt{\frac{13}{9}}$$

Simplify.

$$y-\frac{2}{3}=\frac{\sqrt{13}}{3} y-\frac{2}{3}=-\frac{\sqrt{13}}{3}$$

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

$$y=\frac{\sqrt{13}+2}{3} y=\frac{2-\sqrt{13}}{3}$$