Finite Math Examples

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 u^{2} - 30 u + 45}{125 u + 375}}$

Step 1

Cancel the common factor of $5 u^{2} - 30 u + 45$ and $125 u + 375$ .

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Step 1.1

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

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2}) - 30 u + 45}{125 u + 375}}$

Step 1.2

Factor $5$ out of $- 30 u$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2}) + 5 (- 6 u) + 45}{125 u + 375}}$

Step 1.3

Factor $5$ out of $5 (u^{2}) + 5 (- 6 u)$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u) + 45}{125 u + 375}}$

Step 1.4

Factor $5$ out of $45$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u) + 5 \cdot 9}{125 u + 375}}$

Step 1.5

Factor $5$ out of $5 (u^{2} - 6 u) + 5 \cdot 9$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u + 9)}{125 u + 375}}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $5$ out of $125 u$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u + 9)}{5 (25 u) + 375}}$

Step 1.6.2

Factor $5$ out of $375$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u + 9)}{5 (25 u) + 5 (75)}}$

Step 1.6.3

Factor $5$ out of $5 (25 u) + 5 (75)$ .

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u + 9)}{5 (25 u + 75)}}$

Step 1.6.4

Cancel the common factor.

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{5 (u^{2} - 6 u + 9)}{5 (25 u + 75)}}$

Step 1.6.5

Rewrite the expression.

$\frac{\frac{u^{2} - 9}{25 u + 75}}{\frac{u^{2} - 6 u + 9}{25 u + 75}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{u^{2} - 9}{25 u + 75} \cdot \frac{25 u + 75}{u^{2} - 6 u + 9}$

Step 3

Cancel the common factor of $25 u + 75$ .

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Step 3.1

Cancel the common factor.

$\frac{u^{2} - 9}{25 u + 75} \cdot \frac{25 u + 75}{u^{2} - 6 u + 9}$

Step 3.2

Rewrite the expression.

$(u^{2} - 9) \frac{1}{u^{2} - 6 u + 9}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

$(u^{2} - 9) \frac{1}{u^{2} - 6 u + 3^{2}}$

Step 4.2

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

$6 u = 2 \cdot u \cdot 3$

Step 4.3

Rewrite the polynomial.

$(u^{2} - 9) \frac{1}{u^{2} - 2 \cdot u \cdot 3 + 3^{2}}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 3$ .

$(u^{2} - 9) \frac{1}{{(u - 3)}^{2}}$

Step 5

Multiply $u^{2} - 9$ by $\frac{1}{{(u - 3)}^{2}}$ .

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Cancel the common factor of $u - 3$ and ${(u - 3)}^{2}$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $u - 3$ out of ${(u - 3)}^{2}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{u + 3}{u - 3}$