Solve y+2/y^2-y-6/y^2-1=0

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

Solve for y

$$y=2$$

Steps Using Factoring

Variable y cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by y\left(y-1\right)\left(y+1\right), the least common multiple of y^{2}-y,y^{2}-1.

$$\left(y+1\right)\left(y+2\right)-y\times 6=0$$

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

$$y^{2}+3y+2-y\times 6=0$$

Multiply -1 and 6 to get -6.

$$y^{2}+3y+2-6y=0$$

Combine 3y and -6y to get -3y.

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

To solve the equation, factor y^{2}-3y+2 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=-3 ab=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y-2\right)\left(y-1\right)$$

To find equation solutions, solve y-2=0 and y-1=0.

$$y=2 y=1$$

Variable y cannot be equal to 1.

$$y=2$$

Steps Using Factoring By Grouping

Variable y cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by y\left(y-1\right)\left(y+1\right), the least common multiple of y^{2}-y,y^{2}-1.

$$\left(y+1\right)\left(y+2\right)-y\times 6=0$$

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

$$y^{2}+3y+2-y\times 6=0$$

Multiply -1 and 6 to get -6.

$$y^{2}+3y+2-6y=0$$

Combine 3y and -6y to get -3y.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by+2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite y^{2}-3y+2 as \left(y^{2}-2y\right)+\left(-y+2\right).

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

Factor out y in the first and -1 in the second group.

$$y\left(y-2\right)-\left(y-2\right)$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(y-1\right)$$

To find equation solutions, solve y-2=0 and y-1=0.

$$y=2 y=1$$

Variable y cannot be equal to 1.

$$y=2$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by y\left(y-1\right)\left(y+1\right), the least common multiple of y^{2}-y,y^{2}-1.

$$\left(y+1\right)\left(y+2\right)-y\times 6=0$$

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

$$y^{2}+3y+2-y\times 6=0$$

Multiply -1 and 6 to get -6.

$$y^{2}+3y+2-6y=0$$

Combine 3y and -6y to get -3y.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -3.

$$y=\frac{-\left(-3\right)±\sqrt{9-4\times 2}}{2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-3\right)±\sqrt{9-8}}{2}$$

Add 9 to -8.

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

Take the square root of 1.

$$y=\frac{-\left(-3\right)±1}{2}$$

The opposite of -3 is 3.

$$y=\frac{3±1}{2}$$

Now solve the equation y=\frac{3±1}{2} when ± is plus. Add 3 to 1.

$$y=\frac{4}{2}$$

Divide 4 by 2.

$$y=2$$

Now solve the equation y=\frac{3±1}{2} when ± is minus. Subtract 1 from 3.

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

Divide 2 by 2.

$$y=1$$

The equation is now solved.

$$y=2 y=1$$

Variable y cannot be equal to 1.

$$y=2$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by y\left(y-1\right)\left(y+1\right), the least common multiple of y^{2}-y,y^{2}-1.

$$\left(y+1\right)\left(y+2\right)-y\times 6=0$$

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

$$y^{2}+3y+2-y\times 6=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+3y-y\times 6=-2$$

Multiply -1 and 6 to get -6.

$$y^{2}+3y-6y=-2$$

Combine 3y and -6y to get -3y.

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

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

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

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

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

Add -2 to \frac{9}{4}.

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

Factor y^{2}-3y+\frac{9}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$y=2 y=1$$

Variable y cannot be equal to 1.

$$y=2$$