Finite Math Examples

$\frac{x^{2} - 2 x - 15}{x^{2} + 2 x - 35} \cdot \frac{x^{2} + x - 42}{x + 10 x + 21}$

Step 1

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

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

$- 5, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{x^{2} + 2 x - 35} \cdot \frac{x^{2} + x - 42}{x + 10 x + 21}$

Step 2

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{(x - 5) (x + 7)} \cdot \frac{x^{2} + x - 42}{x + 10 x + 21}$

Step 3

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{(x - 5) (x + 7)} \cdot \frac{(x - 6) (x + 7)}{x + 10 x + 21}$

Step 4

Simplify terms.

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Step 4.1

Add $x$ and $10 x$ .

$\frac{(x - 5) (x + 3)}{(x - 5) (x + 7)} \cdot \frac{(x - 6) (x + 7)}{11 x + 21}$

Step 4.2

Cancel the common factor of $x + 7$ .

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Step 4.2.1

Factor $x + 7$ out of $(x - 5) (x + 7)$ .

$\frac{(x - 5) (x + 3)}{(x + 7) (x - 5)} \cdot \frac{(x - 6) (x + 7)}{11 x + 21}$

Step 4.2.2

Factor $x + 7$ out of $(x - 6) (x + 7)$ .

$\frac{(x - 5) (x + 3)}{(x + 7) (x - 5)} \cdot \frac{(x + 7) (x - 6)}{11 x + 21}$

Step 4.2.3

Cancel the common factor.

$\frac{(x - 5) (x + 3)}{(x + 7) (x - 5)} \cdot \frac{(x + 7) (x - 6)}{11 x + 21}$

Step 4.2.4

Rewrite the expression.

$\frac{(x - 5) (x + 3)}{x - 5} \cdot \frac{x - 6}{11 x + 21}$

Step 4.3

Multiply $\frac{(x - 5) (x + 3)}{x - 5}$ by $\frac{x - 6}{11 x + 21}$ .

$\frac{(x - 5) (x + 3) (x - 6)}{(x - 5) (11 x + 21)}$

Step 4.4

Cancel the common factor of $x - 5$ .

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Step 4.4.1

Cancel the common factor.

$\frac{(x - 5) (x + 3) (x - 6)}{(x - 5) (11 x + 21)}$

Step 4.4.2

Rewrite the expression.

$\frac{(x + 3) (x - 6)}{11 x + 21}$