Basic Math Examples

$\frac{k^{2} + 12 k + 35}{k^{2} + 14 k + 45} \cdot \frac{k^{2} + 9 k}{k^{2} + 10 k + 22}$

Step 1

Factor $k^{2} + 12 k + 35$ 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 $35$ and whose sum is $12$ .

$5, 7$

Step 1.2

Write the factored form using these integers.

$\frac{(k + 5) (k + 7)}{k^{2} + 14 k + 45} \cdot \frac{k^{2} + 9 k}{k^{2} + 10 k + 22}$

Step 2

Factor $k^{2} + 14 k + 45$ 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 $45$ and whose sum is $14$ .

$5, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(k + 5) (k + 7)}{(k + 5) (k + 9)} \cdot \frac{k^{2} + 9 k}{k^{2} + 10 k + 22}$

Step 3

Reduce the expression $\frac{(k + 5) (k + 7)}{(k + 5) (k + 9)}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$\frac{(k + 5) (k + 7)}{(k + 5) (k + 9)} \cdot \frac{k^{2} + 9 k}{k^{2} + 10 k + 22}$

Step 3.2

Rewrite the expression.

$\frac{k + 7}{k + 9} \cdot \frac{k^{2} + 9 k}{k^{2} + 10 k + 22}$

Step 4

Factor $k$ out of $k^{2} + 9 k$ .

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Step 4.1

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

$\frac{k + 7}{k + 9} \cdot \frac{k \cdot k + 9 k}{k^{2} + 10 k + 22}$

Step 4.2

Factor $k$ out of $9 k$ .

$\frac{k + 7}{k + 9} \cdot \frac{k \cdot k + k \cdot 9}{k^{2} + 10 k + 22}$

Step 4.3

Factor $k$ out of $k \cdot k + k \cdot 9$ .

$\frac{k + 7}{k + 9} \cdot \frac{k (k + 9)}{k^{2} + 10 k + 22}$