Finite Math Examples

$\frac{\frac{x^{2} - 9 x + 14}{x^{2} - 6 x + 5}}{\frac{x^{2} - 8 x + 7}{x^{2} - 7 x + 10}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 9 x + 14}{x^{2} - 6 x + 5} \cdot \frac{x^{2} - 7 x + 10}{x^{2} - 8 x + 7}$

Step 2

Factor $x^{2} - 9 x + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 2)}{x^{2} - 6 x + 5} \cdot \frac{x^{2} - 7 x + 10}{x^{2} - 8 x + 7}$

Step 3

Factor $x^{2} - 6 x + 5$ 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{x^{2} - 7 x + 10}{x^{2} - 8 x + 7}$

Step 4

Factor $x^{2} - 7 x + 10$ 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 $- 7$ .

$- 5, - 2$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x^{2} - 8 x + 7}$

Step 5

Factor $x^{2} - 8 x + 7$ 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 $7$ and whose sum is $- 8$ .

$- 7, - 1$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x - 7) (x - 1)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 7$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(x - 7) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x - 7) (x - 1)}$

Step 6.1.2

Rewrite the expression.

$\frac{x - 2}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x - 1}$

Step 6.2

Cancel the common factor of $x - 5$ .

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Step 6.2.1

Cancel the common factor.

$\frac{x - 2}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x - 1}$

Step 6.2.2

Rewrite the expression.

$\frac{x - 2}{x - 1} \cdot \frac{x - 2}{x - 1}$

Step 6.3

Multiply $\frac{x - 2}{x - 1}$ by $\frac{x - 2}{x - 1}$ .

$\frac{(x - 2) (x - 2)}{(x - 1) (x - 1)}$

Step 7

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} (x - 2)}{(x - 1) (x - 1)}$

Step 8

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} {(x - 2)}^{1}}{(x - 1) (x - 1)}$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 2)}^{1 + 1}}{(x - 1) (x - 1)}$

Step 10

Add $1$ and $1$ .

$\frac{{(x - 2)}^{2}}{(x - 1) (x - 1)}$

Step 11

Raise $x - 1$ to the power of $1$ .

$\frac{{(x - 2)}^{2}}{{(x - 1)}^{1} (x - 1)}$

Step 12

Raise $x - 1$ to the power of $1$ .

$\frac{{(x - 2)}^{2}}{{(x - 1)}^{1} {(x - 1)}^{1}}$

Step 13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 2)}^{2}}{{(x - 1)}^{1 + 1}}$

Step 14

Add $1$ and $1$ .

$\frac{{(x - 2)}^{2}}{{(x - 1)}^{2}}$