Finite Math Examples

$\frac{u^{2} - 9 u + 18}{45 - 5 u^{2}}$

Step 1

Factor $u^{2} - 9 u + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 1.2

Write the factored form using these integers.

$\frac{(u - 6) (u - 3)}{45 - 5 u^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $5$ out of $45 - 5 u^{2}$ .

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Step 2.1.1

Factor $5$ out of $45$ .

$\frac{(u - 6) (u - 3)}{5 (9) - 5 u^{2}}$

Step 2.1.2

Factor $5$ out of $- 5 u^{2}$ .

$\frac{(u - 6) (u - 3)}{5 (9) + 5 (- u^{2})}$

Step 2.1.3

Factor $5$ out of $5 (9) + 5 (- u^{2})$ .

$\frac{(u - 6) (u - 3)}{5 (9 - u^{2})}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{(u - 6) (u - 3)}{5 (3^{2} - u^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = u$ .

$\frac{(u - 6) (u - 3)}{5 (3 + u) (3 - u)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $u - 3$ and $3 - u$ .

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Step 3.1.1

Factor $- 1$ out of $u$ .

$\frac{(u - 6) (- 1 (- u) - 3)}{5 (3 + u) (3 - u)}$

Step 3.1.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{(u - 6) (- 1 (- u) - 1 (3))}{5 (3 + u) (3 - u)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- u) - 1 (3)$ .

$\frac{(u - 6) (- 1 (- u + 3))}{5 (3 + u) (3 - u)}$

Step 3.1.4

Reorder terms.

$\frac{(u - 6) (- 1 (- u + 3))}{5 (3 + u) (- u + 3)}$

Step 3.1.5

Cancel the common factor.

$\frac{(u - 6) (- 1 (- u + 3))}{5 (3 + u) (- u + 3)}$

Step 3.1.6

Rewrite the expression.

$\frac{(u - 6) \cdot (- 1)}{5 (3 + u)}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $u - 6$ .

$\frac{- 1 \cdot (u - 6)}{5 (3 + u)}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{u - 6}{5 (3 + u)}$