Basic Math Examples

$\frac{\frac{y^{2} - 2 y - 24}{y^{2} - y - 20}}{\frac{y^{2} + 10 y + 21}{y^{2} - 2 y - 15}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 2 y - 24}{y^{2} - y - 20} \cdot \frac{y^{2} - 2 y - 15}{y^{2} + 10 y + 21}$

Step 2

Factor $y^{2} - 2 y - 24$ using the AC method.

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Step 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 6) (y + 4)}{y^{2} - y - 20} \cdot \frac{y^{2} - 2 y - 15}{y^{2} + 10 y + 21}$

Step 3

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

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

$- 5, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(y - 6) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{y^{2} - 2 y - 15}{y^{2} + 10 y + 21}$

Step 4

Cancel the common factor of $y + 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(y - 6) (y + 4)}{(y - 5) (y + 4)} \cdot \frac{y^{2} - 2 y - 15}{y^{2} + 10 y + 21}$

Step 4.2

Rewrite the expression.

$\frac{y - 6}{y - 5} \cdot \frac{y^{2} - 2 y - 15}{y^{2} + 10 y + 21}$

Step 5

Factor $y^{2} - 2 y - 15$ using the AC method.

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Step 5.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 5.2

Write the factored form using these integers.

$\frac{y - 6}{y - 5} \cdot \frac{(y - 5) (y + 3)}{y^{2} + 10 y + 21}$

Step 6

Factor $y^{2} + 10 y + 21$ using the AC method.

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Step 6.1

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 $21$ and whose sum is $10$ .

$3, 7$

Step 6.2

Write the factored form using these integers.

$\frac{y - 6}{y - 5} \cdot \frac{(y - 5) (y + 3)}{(y + 3) (y + 7)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $y - 5$ .

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Step 7.1.1

Cancel the common factor.

$\frac{y - 6}{y - 5} \cdot \frac{(y - 5) (y + 3)}{(y + 3) (y + 7)}$

Step 7.1.2

Rewrite the expression.

$(y - 6) \frac{y + 3}{(y + 3) (y + 7)}$

Step 7.2

Cancel the common factor of $y + 3$ .

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Step 7.2.1

Cancel the common factor.

$(y - 6) \frac{y + 3}{(y + 3) (y + 7)}$

Step 7.2.2

Rewrite the expression.

$(y - 6) \frac{1}{y + 7}$

Step 7.3

Multiply $y - 6$ by $\frac{1}{y + 7}$ .

$\frac{y - 6}{y + 7}$