Solve 6+5/y-2=4/y+1

$$6+ \frac{ 5 }{ y-2 } = \frac{ 4 }{ y+1 }$$

Solve for y

$$y=\frac{1}{3}\approx 0.333333333$$

$$y=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(y-2\right)\left(y+1\right), the least common multiple of y-2,y+1.

$$\left(y-2\right)\left(y+1\right)\times 6+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y-2 by y+1 and combine like terms.

$$\left(y^{2}-y-2\right)\times 6+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y^{2}-y-2 by 6.

$$6y^{2}-6y-12+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y+1 by 5.

$$6y^{2}-6y-12+5y+5=\left(y-2\right)\times 4$$

Combine -6y and 5y to get -y.

$$6y^{2}-y-12+5=\left(y-2\right)\times 4$$

Add -12 and 5 to get -7.

$$6y^{2}-y-7=\left(y-2\right)\times 4$$

Use the distributive property to multiply y-2 by 4.

$$6y^{2}-y-7=4y-8$$

Subtract 4y from both sides.

$$6y^{2}-y-7-4y=-8$$

Combine -y and -4y to get -5y.

$$6y^{2}-5y-7=-8$$

Add 8 to both sides.

$$6y^{2}-5y-7+8=0$$

Add -7 and 8 to get 1.

$$6y^{2}-5y+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6}}{2\times 6}$$

Square -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\times 6}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-\left(-5\right)±\sqrt{25-24}}{2\times 6}$$

Add 25 to -24.

$$y=\frac{-\left(-5\right)±\sqrt{1}}{2\times 6}$$

Take the square root of 1.

$$y=\frac{-\left(-5\right)±1}{2\times 6}$$

The opposite of -5 is 5.

$$y=\frac{5±1}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{5±1}{12}$$

Now solve the equation y=\frac{5±1}{12} when ± is plus. Add 5 to 1.

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

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation y=\frac{5±1}{12} when ± is minus. Subtract 1 from 5.

$$y=\frac{4}{12}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

$$y=\frac{1}{3}$$

The equation is now solved.

$$y=\frac{1}{2} y=\frac{1}{3}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(y-2\right)\left(y+1\right), the least common multiple of y-2,y+1.

$$\left(y-2\right)\left(y+1\right)\times 6+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y-2 by y+1 and combine like terms.

$$\left(y^{2}-y-2\right)\times 6+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y^{2}-y-2 by 6.

$$6y^{2}-6y-12+\left(y+1\right)\times 5=\left(y-2\right)\times 4$$

Use the distributive property to multiply y+1 by 5.

$$6y^{2}-6y-12+5y+5=\left(y-2\right)\times 4$$

Combine -6y and 5y to get -y.

$$6y^{2}-y-12+5=\left(y-2\right)\times 4$$

Add -12 and 5 to get -7.

$$6y^{2}-y-7=\left(y-2\right)\times 4$$

Use the distributive property to multiply y-2 by 4.

$$6y^{2}-y-7=4y-8$$

Subtract 4y from both sides.

$$6y^{2}-y-7-4y=-8$$

Combine -y and -4y to get -5y.

$$6y^{2}-5y-7=-8$$

Add 7 to both sides.

$$6y^{2}-5y=-8+7$$

Add -8 and 7 to get -1.

$$6y^{2}-5y=-1$$

Divide both sides by 6.

$$\frac{6y^{2}-5y}{6}=-\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}-\frac{5}{6}y=-\frac{1}{6}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{5}{6}y+\left(-\frac{5}{12}\right)^{2}=-\frac{1}{6}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{5}{6}y+\frac{25}{144}=-\frac{1}{6}+\frac{25}{144}$$

Add -\frac{1}{6} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{5}{6}y+\frac{25}{144}=\frac{1}{144}$$

Factor y^{2}-\frac{5}{6}y+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{5}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$y-\frac{5}{12}=\frac{1}{12} y-\frac{5}{12}=-\frac{1}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$y=\frac{1}{2} y=\frac{1}{3}$$