Solve 3y-3/3y-2+6+2y/3y+2=2

$$\frac { 3 y - 3 } { 3 y - 2 } + \frac { 6 + 2 y } { 3 y + 2 } = 2$$

Solve for y

$$y=2$$

$$y = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply 3y+2 by 3y-3 and combine like terms.

$$9y^{2}-3y-6+\left(3y-2\right)\left(6+2y\right)=2\left(3y-2\right)\left(3y+2\right)$$

Use the distributive property to multiply 3y-2 by 6+2y and combine like terms.

$$9y^{2}-3y-6+14y+6y^{2}-12=2\left(3y-2\right)\left(3y+2\right)$$

Combine -3y and 14y to get 11y.

$$9y^{2}+11y-6+6y^{2}-12=2\left(3y-2\right)\left(3y+2\right)$$

Combine 9y^{2} and 6y^{2} to get 15y^{2}.

$$15y^{2}+11y-6-12=2\left(3y-2\right)\left(3y+2\right)$$

Subtract 12 from -6 to get -18.

$$15y^{2}+11y-18=2\left(3y-2\right)\left(3y+2\right)$$

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

$$15y^{2}+11y-18=\left(6y-4\right)\left(3y+2\right)$$

Use the distributive property to multiply 6y-4 by 3y+2 and combine like terms.

$$15y^{2}+11y-18=18y^{2}-8$$

Subtract 18y^{2} from both sides.

$$15y^{2}+11y-18-18y^{2}=-8$$

Combine 15y^{2} and -18y^{2} to get -3y^{2}.

$$-3y^{2}+11y-18=-8$$

Add 8 to both sides.

$$-3y^{2}+11y-18+8=0$$

Add -18 and 8 to get -10.

$$-3y^{2}+11y-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 11 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-11±\sqrt{11^{2}-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Square 11.

$$y=\frac{-11±\sqrt{121-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$y=\frac{-11±\sqrt{121+12\left(-10\right)}}{2\left(-3\right)}$$

Multiply 12 times -10.

$$y=\frac{-11±\sqrt{121-120}}{2\left(-3\right)}$$

Add 121 to -120.

$$y=\frac{-11±\sqrt{1}}{2\left(-3\right)}$$

Take the square root of 1.

$$y=\frac{-11±1}{2\left(-3\right)}$$

Multiply 2 times -3.

$$y=\frac{-11±1}{-6}$$

Now solve the equation y=\frac{-11±1}{-6} when ± is plus. Add -11 to 1.

$$y=-\frac{10}{-6}$$

Reduce the fraction \frac{-10}{-6} to lowest terms by extracting and canceling out 2.

$$y=\frac{5}{3}$$

Now solve the equation y=\frac{-11±1}{-6} when ± is minus. Subtract 1 from -11.

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

Divide -12 by -6.

$$y=2$$

The equation is now solved.

$$y=\frac{5}{3} y=2$$

Steps for Completing the Square

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

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

Use the distributive property to multiply 3y+2 by 3y-3 and combine like terms.

$$9y^{2}-3y-6+\left(3y-2\right)\left(6+2y\right)=2\left(3y-2\right)\left(3y+2\right)$$

Use the distributive property to multiply 3y-2 by 6+2y and combine like terms.

$$9y^{2}-3y-6+14y+6y^{2}-12=2\left(3y-2\right)\left(3y+2\right)$$

Combine -3y and 14y to get 11y.

$$9y^{2}+11y-6+6y^{2}-12=2\left(3y-2\right)\left(3y+2\right)$$

Combine 9y^{2} and 6y^{2} to get 15y^{2}.

$$15y^{2}+11y-6-12=2\left(3y-2\right)\left(3y+2\right)$$

Subtract 12 from -6 to get -18.

$$15y^{2}+11y-18=2\left(3y-2\right)\left(3y+2\right)$$

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

$$15y^{2}+11y-18=\left(6y-4\right)\left(3y+2\right)$$

Use the distributive property to multiply 6y-4 by 3y+2 and combine like terms.

$$15y^{2}+11y-18=18y^{2}-8$$

Subtract 18y^{2} from both sides.

$$15y^{2}+11y-18-18y^{2}=-8$$

Combine 15y^{2} and -18y^{2} to get -3y^{2}.

$$-3y^{2}+11y-18=-8$$

Add 18 to both sides.

$$-3y^{2}+11y=-8+18$$

Add -8 and 18 to get 10.

$$-3y^{2}+11y=10$$

Divide both sides by -3.

$$\frac{-3y^{2}+11y}{-3}=\frac{10}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$y^{2}+\frac{11}{-3}y=\frac{10}{-3}$$

Divide 11 by -3.

$$y^{2}-\frac{11}{3}y=\frac{10}{-3}$$

Divide 10 by -3.

$$y^{2}-\frac{11}{3}y=-\frac{10}{3}$$

Divide -\frac{11}{3}, the coefficient of the x term, by 2 to get -\frac{11}{6}. Then add the square of -\frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{11}{3}y+\left(-\frac{11}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{11}{6}\right)^{2}$$

Square -\frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{11}{3}y+\frac{121}{36}=-\frac{10}{3}+\frac{121}{36}$$

Add -\frac{10}{3} to \frac{121}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{11}{3}y+\frac{121}{36}=\frac{1}{36}$$

Factor y^{2}-\frac{11}{3}y+\frac{121}{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}.

$$\left(y-\frac{11}{6}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{11}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$y-\frac{11}{6}=\frac{1}{6} y-\frac{11}{6}=-\frac{1}{6}$$

Add \frac{11}{6} to both sides of the equation.

$$y=2 y=\frac{5}{3}$$