Basic Math Examples

$\frac{y^{2} + y - 12}{y^{2} - y - 20} \cdot \frac{y^{2} + y - 30}{y^{2} - 2 y} - 3$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} + y - 12$ using the AC method.

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Step 1.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{(y - 3) (y + 4)}{y^{2} - y - 20} \cdot \frac{y^{2} + y - 30}{y^{2} - 2 y} - 3$

Step 1.2

Factor $y^{2} - y - 20$ using the AC method.

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Step 1.2.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{y^{2} + y - 30}{y^{2} - 2 y} - 3$

Step 1.3

Factor $y^{2} + y - 30$ using the AC method.

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Step 1.3.1

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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 1.3.2

Write the factored form using these integers.

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{(y - 5) (y + 6)}{y^{2} - 2 y} - 3$

Step 1.4

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

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Step 1.4.1

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

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{(y - 5) (y + 6)}{y \cdot y - 2 y} - 3$

Step 1.4.2

Factor $y$ out of $- 2 y$ .

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{(y - 5) (y + 6)}{y \cdot y + y \cdot - 2} - 3$

Step 1.4.3

Factor $y$ out of $y \cdot y + y \cdot - 2$ .

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{(y - 5) (y + 6)}{y (y - 2)} - 3$

Step 1.5

Cancel the common factor of $y - 5$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(y - 3) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{(y - 5) (y + 6)}{y (y - 2)} - 3$

Step 1.5.2

Rewrite the expression.

$\frac{(y - 3) (y + 4)}{y + 4} \cdot \frac{y + 6}{y (y - 2)} - 3$

Step 1.6

Multiply $\frac{(y - 3) (y + 4)}{y + 4}$ by $\frac{y + 6}{y (y - 2)}$ .

$\frac{(y - 3) (y + 4) (y + 6)}{(y + 4) (y (y - 2))} - 3$

Step 1.7

Cancel the common factor of $y + 4$ .

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Step 1.7.1

Cancel the common factor.

$\frac{(y - 3) (y + 4) (y + 6)}{(y + 4) (y (y - 2))} - 3$

Step 1.7.2

Rewrite the expression.

$\frac{(y - 3) (y + 6)}{y (y - 2)} - 3$

Step 2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{y (y - 2)}{y (y - 2)}$ .

$\frac{(y - 3) (y + 6)}{y (y - 2)} - 3 \cdot \frac{y (y - 2)}{y (y - 2)}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 3$ and $\frac{y (y - 2)}{y (y - 2)}$ .

$\frac{(y - 3) (y + 6)}{y (y - 2)} + \frac{- 3 (y (y - 2))}{y (y - 2)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(y - 3) (y + 6) - 3 (y (y - 2))}{y (y - 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(y - 3) (y + 6)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{y (y + 6) - 3 (y + 6) - 3 y (y - 2)}{y (y - 2)}$

Step 4.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 6 - 3 (y + 6) - 3 y (y - 2)}{y (y - 2)}$

Step 4.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 6 - 3 y - 3 \cdot 6 - 3 y (y - 2)}{y (y - 2)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot 6 - 3 y - 3 \cdot 6 - 3 y (y - 2)}{y (y - 2)}$

Step 4.2.1.2

Move $6$ to the left of $y$ .

$\frac{y^{2} + 6 \cdot y - 3 y - 3 \cdot 6 - 3 y (y - 2)}{y (y - 2)}$

Step 4.2.1.3

Multiply $- 3$ by $6$ .

$\frac{y^{2} + 6 y - 3 y - 18 - 3 y (y - 2)}{y (y - 2)}$

Step 4.2.2

Subtract $3 y$ from $6 y$ .

$\frac{y^{2} + 3 y - 18 - 3 y (y - 2)}{y (y - 2)}$

Step 4.3

Apply the distributive property.

$\frac{y^{2} + 3 y - 18 - 3 y \cdot y - 3 y \cdot - 2}{y (y - 2)}$

Step 4.4

Multiply $y$ by $y$ by adding the exponents.

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Step 4.4.1

Move $y$ .

$\frac{y^{2} + 3 y - 18 - 3 (y \cdot y) - 3 y \cdot - 2}{y (y - 2)}$

Step 4.4.2

Multiply $y$ by $y$ .

$\frac{y^{2} + 3 y - 18 - 3 y^{2} - 3 y \cdot - 2}{y (y - 2)}$

Step 4.5

Multiply $- 2$ by $- 3$ .

$\frac{y^{2} + 3 y - 18 - 3 y^{2} + 6 y}{y (y - 2)}$

Step 4.6

Subtract $3 y^{2}$ from $y^{2}$ .

$\frac{- 2 y^{2} + 3 y - 18 + 6 y}{y (y - 2)}$

Step 4.7

Add $3 y$ and $6 y$ .

$\frac{- 2 y^{2} + 9 y - 18}{y (y - 2)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 2 y^{2}$ .

$\frac{- (2 y^{2}) + 9 y - 18}{y (y - 2)}$

Step 5.2

Factor $- 1$ out of $9 y$ .

$\frac{- (2 y^{2}) - (- 9 y) - 18}{y (y - 2)}$

Step 5.3

Factor $- 1$ out of $- (2 y^{2}) - (- 9 y)$ .

$\frac{- (2 y^{2} - 9 y) - 18}{y (y - 2)}$

Step 5.4

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- (2 y^{2} - 9 y) - 1 (18)}{y (y - 2)}$

Step 5.5

Factor $- 1$ out of $- (2 y^{2} - 9 y) - 1 (18)$ .

$\frac{- (2 y^{2} - 9 y + 18)}{y (y - 2)}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (2 y^{2} - 9 y + 18)$ as $- 1 (2 y^{2} - 9 y + 18)$ .

$\frac{- 1 (2 y^{2} - 9 y + 18)}{y (y - 2)}$

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{2 y^{2} - 9 y + 18}{y (y - 2)}$