Algebra Examples

$\frac{5 y^{2}}{5 y^{2} + 30 y - 35} \cdot \frac{2 y^{3} + 30 y^{2} + 112 y}{8 y + 44}$

Step 1

Set the denominator in $\frac{5 y^{2}}{5 y^{2} + 30 y - 35}$ equal to $0$ to find where the expression is undefined.

$5 y^{2} + 30 y - 35 = 0$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Factor the left side of the equation.

Tap for more steps...

Step 2.1.1

Factor $5$ out of $5 y^{2} + 30 y - 35$ .

Tap for more steps...

Step 2.1.1.1

Factor $5$ out of $5 y^{2}$ .

$5 (y^{2}) + 30 y - 35 = 0$

Step 2.1.1.2

Factor $5$ out of $30 y$ .

$5 (y^{2}) + 5 (6 y) - 35 = 0$

Step 2.1.1.3

Factor $5$ out of $- 35$ .

$5 y^{2} + 5 (6 y) + 5 \cdot - 7 = 0$

Step 2.1.1.4

Factor $5$ out of $5 y^{2} + 5 (6 y)$ .

$5 (y^{2} + 6 y) + 5 \cdot - 7 = 0$

Step 2.1.1.5

Factor $5$ out of $5 (y^{2} + 6 y) + 5 \cdot - 7$ .

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

Step 2.1.2

Factor.

Tap for more steps...

Step 2.1.2.1

Factor $y^{2} + 6 y - 7$ using the AC method.

Tap for more steps...

Step 2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 2.1.2.1.2

Write the factored form using these integers.

$5 ((y - 1) (y + 7)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$5 (y - 1) (y + 7) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 1 = 0$

$y + 7 = 0$

Step 2.3

Set $y - 1$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.3.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 1$

Step 2.4

Set $y + 7$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.4.1

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

Step 2.4.2

Subtract $7$ from both sides of the equation.

$y = - 7$

Step 2.5

The final solution is all the values that make $5 (y - 1) (y + 7) = 0$ true.

$y = 1, - 7$

Step 3

Set the denominator in $\frac{2 y^{3} + 30 y^{2} + 112 y}{8 y + 44}$ equal to $0$ to find where the expression is undefined.

$8 y + 44 = 0$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Subtract $44$ from both sides of the equation.

$8 y = - 44$

Step 4.2

Divide each term in $8 y = - 44$ by $8$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $8 y = - 44$ by $8$ .

$\frac{8 y}{8} = \frac{- 44}{8}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 44}{8}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 44}{8}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Cancel the common factor of $- 44$ and $8$ .

Tap for more steps...

Step 4.2.3.1.1

Factor $4$ out of $- 44$ .

$y = \frac{4 (- 11)}{8}$

Step 4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.1.2.1

Factor $4$ out of $8$ .

$y = \frac{4 \cdot - 11}{4 \cdot 2}$

Step 4.2.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot - 11}{4 \cdot 2}$

Step 4.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 11}{2}$

Step 4.2.3.2

Move the negative in front of the fraction.

$y = - \frac{11}{2}$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$y = - 7, y = - \frac{11}{2}, y = 1$

Step 6