Solve y+2/y-2-2/y+2=-7/3

$$\frac { y + 2 } { y - 2 } - \frac { 2 } { y + 2 } = - \frac { 7 } { 3 }$$

Solve for y

$$y=-1$$

$$y=\frac{2}{5}=0.4$$

Steps Using the Quadratic Formula

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

$$\left(3y+6\right)\left(y+2\right)-\left(3y-6\right)\times 2=-7\left(y-2\right)\left(y+2\right)$$

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

$$3y^{2}+12y+12-\left(3y-6\right)\times 2=-7\left(y-2\right)\left(y+2\right)$$

Use the distributive property to multiply 3y-6 by 2.

$$3y^{2}+12y+12-\left(6y-12\right)=-7\left(y-2\right)\left(y+2\right)$$

To find the opposite of 6y-12, find the opposite of each term.

$$3y^{2}+12y+12-6y+12=-7\left(y-2\right)\left(y+2\right)$$

Combine 12y and -6y to get 6y.

$$3y^{2}+6y+12+12=-7\left(y-2\right)\left(y+2\right)$$

Add 12 and 12 to get 24.

$$3y^{2}+6y+24=-7\left(y-2\right)\left(y+2\right)$$

Use the distributive property to multiply -7 by y-2.

$$3y^{2}+6y+24=\left(-7y+14\right)\left(y+2\right)$$

Use the distributive property to multiply -7y+14 by y+2 and combine like terms.

$$3y^{2}+6y+24=-7y^{2}+28$$

Add 7y^{2} to both sides.

$$3y^{2}+6y+24+7y^{2}=28$$

Combine 3y^{2} and 7y^{2} to get 10y^{2}.

$$10y^{2}+6y+24=28$$

Subtract 28 from both sides.

$$10y^{2}+6y+24-28=0$$

Subtract 28 from 24 to get -4.

$$10y^{2}+6y-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 6 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\times 10\left(-4\right)}}{2\times 10}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\times 10\left(-4\right)}}{2\times 10}$$

Multiply -4 times 10.

$$y=\frac{-6±\sqrt{36-40\left(-4\right)}}{2\times 10}$$

Multiply -40 times -4.

$$y=\frac{-6±\sqrt{36+160}}{2\times 10}$$

Add 36 to 160.

$$y=\frac{-6±\sqrt{196}}{2\times 10}$$

Take the square root of 196.

$$y=\frac{-6±14}{2\times 10}$$

Multiply 2 times 10.

$$y=\frac{-6±14}{20}$$

Now solve the equation y=\frac{-6±14}{20} when ± is plus. Add -6 to 14.

$$y=\frac{8}{20}$$

Reduce the fraction \frac{8}{20} to lowest terms by extracting and canceling out 4.

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

Now solve the equation y=\frac{-6±14}{20} when ± is minus. Subtract 14 from -6.

$$y=-\frac{20}{20}$$

Divide -20 by 20.

$$y=-1$$

The equation is now solved.

$$y=\frac{2}{5} y=-1$$

Steps for Completing the Square

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

$$\left(3y+6\right)\left(y+2\right)-\left(3y-6\right)\times 2=-7\left(y-2\right)\left(y+2\right)$$

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

$$3y^{2}+12y+12-\left(3y-6\right)\times 2=-7\left(y-2\right)\left(y+2\right)$$

Use the distributive property to multiply 3y-6 by 2.

$$3y^{2}+12y+12-\left(6y-12\right)=-7\left(y-2\right)\left(y+2\right)$$

To find the opposite of 6y-12, find the opposite of each term.

$$3y^{2}+12y+12-6y+12=-7\left(y-2\right)\left(y+2\right)$$

Combine 12y and -6y to get 6y.

$$3y^{2}+6y+12+12=-7\left(y-2\right)\left(y+2\right)$$

Add 12 and 12 to get 24.

$$3y^{2}+6y+24=-7\left(y-2\right)\left(y+2\right)$$

Use the distributive property to multiply -7 by y-2.

$$3y^{2}+6y+24=\left(-7y+14\right)\left(y+2\right)$$

Use the distributive property to multiply -7y+14 by y+2 and combine like terms.

$$3y^{2}+6y+24=-7y^{2}+28$$

Add 7y^{2} to both sides.

$$3y^{2}+6y+24+7y^{2}=28$$

Combine 3y^{2} and 7y^{2} to get 10y^{2}.

$$10y^{2}+6y+24=28$$

Subtract 24 from both sides.

$$10y^{2}+6y=28-24$$

Subtract 24 from 28 to get 4.

$$10y^{2}+6y=4$$

Divide both sides by 10.

$$\frac{10y^{2}+6y}{10}=\frac{4}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$y^{2}+\frac{6}{10}y=\frac{4}{10}$$

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

$$y^{2}+\frac{3}{5}y=\frac{4}{10}$$

Reduce the fraction \frac{4}{10} to lowest terms by extracting and canceling out 2.

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

Divide \frac{3}{5}, the coefficient of the x term, by 2 to get \frac{3}{10}. Then add the square of \frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{3}{5}y+\left(\frac{3}{10}\right)^{2}=\frac{2}{5}+\left(\frac{3}{10}\right)^{2}$$

Square \frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{3}{5}y+\frac{9}{100}=\frac{2}{5}+\frac{9}{100}$$

Add \frac{2}{5} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{3}{5}y+\frac{9}{100}=\frac{49}{100}$$

Factor y^{2}+\frac{3}{5}y+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{3}{10}\right)^{2}=\frac{49}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{3}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Simplify.

$$y+\frac{3}{10}=\frac{7}{10} y+\frac{3}{10}=-\frac{7}{10}$$

Subtract \frac{3}{10} from both sides of the equation.

$$y=\frac{2}{5} y=-1$$