Finite Math Examples

$\frac{9 x^{2} + 45 x + 54}{3 x^{2} + 15 + 18}$

Step 1

Cancel the common factor of $9 x^{2} + 45 x + 54$ and $3 x^{2} + 15 + 18$ .

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Step 1.1

Factor $3$ out of $9 x^{2}$ .

$\frac{3 (3 x^{2}) + 45 x + 54}{3 x^{2} + 15 + 18}$

Step 1.2

Factor $3$ out of $45 x$ .

$\frac{3 (3 x^{2}) + 3 (15 x) + 54}{3 x^{2} + 15 + 18}$

Step 1.3

Factor $3$ out of $3 (3 x^{2}) + 3 (15 x)$ .

$\frac{3 (3 x^{2} + 15 x) + 54}{3 x^{2} + 15 + 18}$

Step 1.4

Factor $3$ out of $54$ .

$\frac{3 (3 x^{2} + 15 x) + 3 \cdot 18}{3 x^{2} + 15 + 18}$

Step 1.5

Factor $3$ out of $3 (3 x^{2} + 15 x) + 3 (18)$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 x^{2} + 15 + 18}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 (x^{2}) + 15 + 18}$

Step 1.6.2

Factor $3$ out of $15$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 x^{2} + 3 \cdot 5 + 18}$

Step 1.6.3

Factor $3$ out of $3 x^{2} + 3 \cdot 5$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 (x^{2} + 5) + 18}$

Step 1.6.4

Factor $3$ out of $18$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 (x^{2} + 5) + 3 (6)}$

Step 1.6.5

Factor $3$ out of $3 (x^{2} + 5) + 3 (6)$ .

$\frac{3 (3 x^{2} + 15 x + 18)}{3 (x^{2} + 5 + 6)}$

Step 1.6.6

Cancel the common factor.

$\frac{3 (3 x^{2} + 15 x + 18)}{3 (x^{2} + 5 + 6)}$

Step 1.6.7

Rewrite the expression.

$\frac{3 x^{2} + 15 x + 18}{x^{2} + 5 + 6}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 x^{2} + 15 x + 18$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) + 15 x + 18}{x^{2} + 5 + 6}$

Step 2.1.2

Factor $3$ out of $15 x$ .

$\frac{3 (x^{2}) + 3 (5 x) + 18}{x^{2} + 5 + 6}$

Step 2.1.3

Factor $3$ out of $18$ .

$\frac{3 x^{2} + 3 (5 x) + 3 \cdot 6}{x^{2} + 5 + 6}$

Step 2.1.4

Factor $3$ out of $3 x^{2} + 3 (5 x)$ .

$\frac{3 (x^{2} + 5 x) + 3 \cdot 6}{x^{2} + 5 + 6}$

Step 2.1.5

Factor $3$ out of $3 (x^{2} + 5 x) + 3 \cdot 6$ .

$\frac{3 (x^{2} + 5 x + 6)}{x^{2} + 5 + 6}$

Step 2.2

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 2.2.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{3 ((x + 2) (x + 3))}{x^{2} + 5 + 6}$

$\frac{3 (x + 2) (x + 3)}{x^{2} + 5 + 6}$

Step 3

Add $5$ and $6$ .

$\frac{3 (x + 2) (x + 3)}{x^{2} + 11}$