Finite Math Examples

$(\frac{k^{2} + 11 k + 18}{k^{2} + 15 k + 54}) (\frac{k^{2} + 6 k}{k^{2} \cdot (7 k) + 10})$

Step 1

Factor $k^{2} + 11 k + 18$ 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 $11$ .

$2, 9$

Step 1.2

Write the factored form using these integers.

$\frac{(k + 2) (k + 9)}{k^{2} + 15 k + 54} \cdot \frac{k^{2} + 6 k}{k^{2} \cdot (7 k) + 10}$

Step 2

Factor $k^{2} + 15 k + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(k + 2) (k + 9)}{(k + 6) (k + 9)} \cdot \frac{k^{2} + 6 k}{k^{2} \cdot (7 k) + 10}$

Step 3

Reduce the expression $\frac{(k + 2) (k + 9)}{(k + 6) (k + 9)}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$\frac{(k + 2) (k + 9)}{(k + 6) (k + 9)} \cdot \frac{k^{2} + 6 k}{k^{2} \cdot (7 k) + 10}$

Step 3.2

Rewrite the expression.

$\frac{k + 2}{k + 6} \cdot \frac{k^{2} + 6 k}{k^{2} \cdot (7 k) + 10}$

Step 4

Factor $k$ out of $k^{2} + 6 k$ .

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Step 4.1

Factor $k$ out of $k^{2}$ .

$\frac{k + 2}{k + 6} \cdot \frac{k \cdot k + 6 k}{k^{2} \cdot (7 k) + 10}$

Step 4.2

Factor $k$ out of $6 k$ .

$\frac{k + 2}{k + 6} \cdot \frac{k \cdot k + k \cdot 6}{k^{2} \cdot (7 k) + 10}$

Step 4.3

Factor $k$ out of $k \cdot k + k \cdot 6$ .

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{k^{2} \cdot (7 k) + 10}$

Step 5

Rewrite using the commutative property of multiplication.

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{7 k^{2} k + 10}$

Step 6

Multiply $k^{2}$ by $k$ by adding the exponents.

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Step 6.1

Move $k$ .

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{7 (k \cdot k^{2}) + 10}$

Step 6.2

Multiply $k$ by $k^{2}$ .

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Step 6.2.1

Raise $k$ to the power of $1$ .

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{7 (k^{1} k^{2}) + 10}$

Step 6.2.2

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

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{7 k^{1 + 2} + 10}$

Step 6.3

Add $1$ and $2$ .

$\frac{k + 2}{k + 6} \cdot \frac{k (k + 6)}{7 k^{3} + 10}$