Solve 2=y/y+3-3/y-5

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

Solve for y

$$y=-7$$

$$y=3$$

Steps Using the Quadratic Formula

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

$$2\left(y-5\right)\left(y+3\right)=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply 2 by y-5.

$$\left(2y-10\right)\left(y+3\right)=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply 2y-10 by y+3 and combine like terms.

$$2y^{2}-4y-30=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply y-5 by y.

$$2y^{2}-4y-30=y^{2}-5y-\left(y+3\right)\times 3$$

Use the distributive property to multiply y+3 by 3.

$$2y^{2}-4y-30=y^{2}-5y-\left(3y+9\right)$$

To find the opposite of 3y+9, find the opposite of each term.

$$2y^{2}-4y-30=y^{2}-5y-3y-9$$

Combine -5y and -3y to get -8y.

$$2y^{2}-4y-30=y^{2}-8y-9$$

Subtract y^{2} from both sides.

$$2y^{2}-4y-30-y^{2}=-8y-9$$

Combine 2y^{2} and -y^{2} to get y^{2}.

$$y^{2}-4y-30=-8y-9$$

Add 8y to both sides.

$$y^{2}-4y-30+8y=-9$$

Combine -4y and 8y to get 4y.

$$y^{2}+4y-30=-9$$

Add 9 to both sides.

$$y^{2}+4y-30+9=0$$

Add -30 and 9 to get -21.

$$y^{2}+4y-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-4±\sqrt{4^{2}-4\left(-21\right)}}{2}$$

Square 4.

$$y=\frac{-4±\sqrt{16-4\left(-21\right)}}{2}$$

Multiply -4 times -21.

$$y=\frac{-4±\sqrt{16+84}}{2}$$

Add 16 to 84.

$$y=\frac{-4±\sqrt{100}}{2}$$

Take the square root of 100.

$$y=\frac{-4±10}{2}$$

Now solve the equation y=\frac{-4±10}{2} when ± is plus. Add -4 to 10.

$$y=\frac{6}{2}$$

Divide 6 by 2.

$$y=3$$

Now solve the equation y=\frac{-4±10}{2} when ± is minus. Subtract 10 from -4.

$$y=-\frac{14}{2}$$

Divide -14 by 2.

$$y=-7$$

The equation is now solved.

$$y=3 y=-7$$

Steps for Completing the Square

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

$$2\left(y-5\right)\left(y+3\right)=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply 2 by y-5.

$$\left(2y-10\right)\left(y+3\right)=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply 2y-10 by y+3 and combine like terms.

$$2y^{2}-4y-30=\left(y-5\right)y-\left(y+3\right)\times 3$$

Use the distributive property to multiply y-5 by y.

$$2y^{2}-4y-30=y^{2}-5y-\left(y+3\right)\times 3$$

Use the distributive property to multiply y+3 by 3.

$$2y^{2}-4y-30=y^{2}-5y-\left(3y+9\right)$$

To find the opposite of 3y+9, find the opposite of each term.

$$2y^{2}-4y-30=y^{2}-5y-3y-9$$

Combine -5y and -3y to get -8y.

$$2y^{2}-4y-30=y^{2}-8y-9$$

Subtract y^{2} from both sides.

$$2y^{2}-4y-30-y^{2}=-8y-9$$

Combine 2y^{2} and -y^{2} to get y^{2}.

$$y^{2}-4y-30=-8y-9$$

Add 8y to both sides.

$$y^{2}-4y-30+8y=-9$$

Combine -4y and 8y to get 4y.

$$y^{2}+4y-30=-9$$

Add 30 to both sides.

$$y^{2}+4y=-9+30$$

Add -9 and 30 to get 21.

$$y^{2}+4y=21$$

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

$$y^{2}+4y+2^{2}=21+2^{2}$$

Square 2.

$$y^{2}+4y+4=21+4$$

Add 21 to 4.

$$y^{2}+4y+4=25$$

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

$$\left(y+2\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+2\right)^{2}}=\sqrt{25}$$

Simplify.

$$y+2=5 y+2=-5$$

Subtract 2 from both sides of the equation.

$$y=3 y=-7$$