Finite Math Examples

$\frac{\frac{x^{2} - 16}{x^{2} - 10 x + 25}}{\frac{3 x - 12}{x^{2} - 3 x - 10}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 16}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2} - 4^{2}}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{(x + 4) (x - 4)}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x + 4) (x - 4)}{x^{2} - 10 x + 5^{2}} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 3.3

Rewrite the polynomial.

$\frac{(x + 4) (x - 4)}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{(x + 4) (x - 4)}{{(x - 5)}^{2}} \cdot \frac{x^{2} - 3 x - 10}{3 x - 12}$

Step 4

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 4.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{{(x - 5)}^{2}} \cdot \frac{(x - 5) (x + 2)}{3 x - 12}$

Step 5

Simplify terms.

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Step 5.1

Factor $3$ out of $3 x - 12$ .

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Step 5.1.1

Factor $3$ out of $3 x$ .

$\frac{(x + 4) (x - 4)}{{(x - 5)}^{2}} \cdot \frac{(x - 5) (x + 2)}{3 (x) - 12}$

Step 5.1.2

Factor $3$ out of $- 12$ .

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

Step 5.1.3

Factor $3$ out of $3 x + 3 \cdot - 4$ .

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

Step 5.2

Combine.

$\frac{(x + 4) (x - 4) ((x - 5) (x + 2))}{{(x - 5)}^{2} (3 (x - 4))}$

Step 5.3

Cancel the common factor of $x - 4$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(x + 4) (x - 4) ((x - 5) (x + 2))}{{(x - 5)}^{2} (3 (x - 4))}$

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of $x - 5$ and ${(x - 5)}^{2}$ .

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Step 5.4.1

Factor $x - 5$ out of $(x + 4) ((x - 5) (x + 2))$ .

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

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $x - 5$ out of ${(x - 5)}^{2} \cdot (3)$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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

Step 5.5

Move $3$ to the left of $x - 5$ .

$\frac{(x + 4) (x + 2)}{3 (x - 5)}$