Solve 18y+7/y^3-1=30/y^2-1-13/y^2+y+1

$$\frac { 18 y + 7 } { y ^ { 3 } - 1 } = \frac { 30 } { y ^ { 2 } - 1 } - \frac { 13 } { y ^ { 2 } + y + 1 }$$

Solve for y

$$y=-4$$

$$y=9$$

Steps Using Factoring

Variable y cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(y-1\right)\left(y+1\right)\left(y^{2}+y+1\right), the least common multiple of y^{3}-1,y^{2}-1,y^{2}+y+1.

$$\left(y+1\right)\left(18y+7\right)=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y+1 by 18y+7 and combine like terms.

$$18y^{2}+25y+7=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}+y+1 by 30.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}-1 by 13.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(13y^{2}-13\right)$$

To find the opposite of 13y^{2}-13, find the opposite of each term.

$$18y^{2}+25y+7=30y^{2}+30y+30-13y^{2}+13$$

Combine 30y^{2} and -13y^{2} to get 17y^{2}.

$$18y^{2}+25y+7=17y^{2}+30y+30+13$$

Add 30 and 13 to get 43.

$$18y^{2}+25y+7=17y^{2}+30y+43$$

Subtract 17y^{2} from both sides.

$$18y^{2}+25y+7-17y^{2}=30y+43$$

Combine 18y^{2} and -17y^{2} to get y^{2}.

$$y^{2}+25y+7=30y+43$$

Subtract 30y from both sides.

$$y^{2}+25y+7-30y=43$$

Combine 25y and -30y to get -5y.

$$y^{2}-5y+7=43$$

Subtract 43 from both sides.

$$y^{2}-5y+7-43=0$$

Subtract 43 from 7 to get -36.

$$y^{2}-5y-36=0$$

To solve the equation, factor y^{2}-5y-36 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.

$$a+b=-5 ab=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

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

$$\left(y-9\right)\left(y+4\right)$$

To find equation solutions, solve y-9=0 and y+4=0.

$$y=9 y=-4$$

Steps Using Factoring By Grouping

Variable y cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(y-1\right)\left(y+1\right)\left(y^{2}+y+1\right), the least common multiple of y^{3}-1,y^{2}-1,y^{2}+y+1.

$$\left(y+1\right)\left(18y+7\right)=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y+1 by 18y+7 and combine like terms.

$$18y^{2}+25y+7=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}+y+1 by 30.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}-1 by 13.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(13y^{2}-13\right)$$

To find the opposite of 13y^{2}-13, find the opposite of each term.

$$18y^{2}+25y+7=30y^{2}+30y+30-13y^{2}+13$$

Combine 30y^{2} and -13y^{2} to get 17y^{2}.

$$18y^{2}+25y+7=17y^{2}+30y+30+13$$

Add 30 and 13 to get 43.

$$18y^{2}+25y+7=17y^{2}+30y+43$$

Subtract 17y^{2} from both sides.

$$18y^{2}+25y+7-17y^{2}=30y+43$$

Combine 18y^{2} and -17y^{2} to get y^{2}.

$$y^{2}+25y+7=30y+43$$

Subtract 30y from both sides.

$$y^{2}+25y+7-30y=43$$

Combine 25y and -30y to get -5y.

$$y^{2}-5y+7=43$$

Subtract 43 from both sides.

$$y^{2}-5y+7-43=0$$

Subtract 43 from 7 to get -36.

$$y^{2}-5y-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-36. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\left(-36\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

Rewrite y^{2}-5y-36 as \left(y^{2}-9y\right)+\left(4y-36\right).

$$\left(y^{2}-9y\right)+\left(4y-36\right)$$

Factor out y in the first and 4 in the second group.

$$y\left(y-9\right)+4\left(y-9\right)$$

Factor out common term y-9 by using distributive property.

$$\left(y-9\right)\left(y+4\right)$$

To find equation solutions, solve y-9=0 and y+4=0.

$$y=9 y=-4$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(y-1\right)\left(y+1\right)\left(y^{2}+y+1\right), the least common multiple of y^{3}-1,y^{2}-1,y^{2}+y+1.

$$\left(y+1\right)\left(18y+7\right)=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y+1 by 18y+7 and combine like terms.

$$18y^{2}+25y+7=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}+y+1 by 30.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}-1 by 13.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(13y^{2}-13\right)$$

To find the opposite of 13y^{2}-13, find the opposite of each term.

$$18y^{2}+25y+7=30y^{2}+30y+30-13y^{2}+13$$

Combine 30y^{2} and -13y^{2} to get 17y^{2}.

$$18y^{2}+25y+7=17y^{2}+30y+30+13$$

Add 30 and 13 to get 43.

$$18y^{2}+25y+7=17y^{2}+30y+43$$

Subtract 17y^{2} from both sides.

$$18y^{2}+25y+7-17y^{2}=30y+43$$

Combine 18y^{2} and -17y^{2} to get y^{2}.

$$y^{2}+25y+7=30y+43$$

Subtract 30y from both sides.

$$y^{2}+25y+7-30y=43$$

Combine 25y and -30y to get -5y.

$$y^{2}-5y+7=43$$

Subtract 43 from both sides.

$$y^{2}-5y+7-43=0$$

Subtract 43 from 7 to get -36.

$$y^{2}-5y-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-36\right)}}{2}$$

Square -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$y=\frac{-\left(-5\right)±\sqrt{25+144}}{2}$$

Add 25 to 144.

$$y=\frac{-\left(-5\right)±\sqrt{169}}{2}$$

Take the square root of 169.

$$y=\frac{-\left(-5\right)±13}{2}$$

The opposite of -5 is 5.

$$y=\frac{5±13}{2}$$

Now solve the equation y=\frac{5±13}{2} when ± is plus. Add 5 to 13.

$$y=\frac{18}{2}$$

Divide 18 by 2.

$$y=9$$

Now solve the equation y=\frac{5±13}{2} when ± is minus. Subtract 13 from 5.

$$y=-\frac{8}{2}$$

Divide -8 by 2.

$$y=-4$$

The equation is now solved.

$$y=9 y=-4$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(y-1\right)\left(y+1\right)\left(y^{2}+y+1\right), the least common multiple of y^{3}-1,y^{2}-1,y^{2}+y+1.

$$\left(y+1\right)\left(18y+7\right)=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y+1 by 18y+7 and combine like terms.

$$18y^{2}+25y+7=\left(y^{2}+y+1\right)\times 30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}+y+1 by 30.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(y^{2}-1\right)\times 13$$

Use the distributive property to multiply y^{2}-1 by 13.

$$18y^{2}+25y+7=30y^{2}+30y+30-\left(13y^{2}-13\right)$$

To find the opposite of 13y^{2}-13, find the opposite of each term.

$$18y^{2}+25y+7=30y^{2}+30y+30-13y^{2}+13$$

Combine 30y^{2} and -13y^{2} to get 17y^{2}.

$$18y^{2}+25y+7=17y^{2}+30y+30+13$$

Add 30 and 13 to get 43.

$$18y^{2}+25y+7=17y^{2}+30y+43$$

Subtract 17y^{2} from both sides.

$$18y^{2}+25y+7-17y^{2}=30y+43$$

Combine 18y^{2} and -17y^{2} to get y^{2}.

$$y^{2}+25y+7=30y+43$$

Subtract 30y from both sides.

$$y^{2}+25y+7-30y=43$$

Combine 25y and -30y to get -5y.

$$y^{2}-5y+7=43$$

Subtract 7 from both sides.

$$y^{2}-5y=43-7$$

Subtract 7 from 43 to get 36.

$$y^{2}-5y=36$$

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

$$y^{2}-5y+\left(-\frac{5}{2}\right)^{2}=36+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-5y+\frac{25}{4}=36+\frac{25}{4}$$

Add 36 to \frac{25}{4}.

$$y^{2}-5y+\frac{25}{4}=\frac{169}{4}$$

Factor y^{2}-5y+\frac{25}{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}.

$$\left(y-\frac{5}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$y-\frac{5}{2}=\frac{13}{2} y-\frac{5}{2}=-\frac{13}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$y=9 y=-4$$