Finite Math Examples

$\frac{x^{2} - 38 x + 361}{x^{2} + 16 x - 57}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $361$ as $19^{2}$ .

$\frac{x^{2} - 38 x + 19^{2}}{x^{2} + 16 x - 57}$

Step 1.2

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

$38 x = 2 \cdot x \cdot 19$

Step 1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 19 + 19^{2}}{x^{2} + 16 x - 57}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 19$ .

$\frac{{(x - 19)}^{2}}{x^{2} + 16 x - 57}$

Step 2

Factor $x^{2} + 16 x - 57$ 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 $- 57$ and whose sum is $16$ .

$- 3, 19$

Step 2.2

Write the factored form using these integers.

$\frac{{(x - 19)}^{2}}{(x - 3) (x + 19)}$