Basic Math Examples

$\frac{k^{2} - 6 k - 7}{k^{2} + k} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 1

Factor $k^{2} - 6 k - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 1.2

Write the factored form using these integers.

$\frac{(k - 7) (k + 1)}{k^{2} + k} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 2

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

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Step 2.1

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

$\frac{(k - 7) (k + 1)}{k \cdot k + k} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 2.2

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

$\frac{(k - 7) (k + 1)}{k \cdot k + k^{1}} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 2.3

Factor $k$ out of $k^{1}$ .

$\frac{(k - 7) (k + 1)}{k \cdot k + k \cdot 1} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 2.4

Factor $k$ out of $k \cdot k + k \cdot 1$ .

$\frac{(k - 7) (k + 1)}{k (k + 1)} \cdot \frac{k^{2} + 5 k + 6}{k^{2} - 5 k - 14}$

Step 3

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

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Step 3.1

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 3.2

Write the factored form using these integers.

$\frac{(k - 7) (k + 1)}{k (k + 1)} \cdot \frac{(k + 2) (k + 3)}{k^{2} - 5 k - 14}$

Step 4

Factor $k^{2} - 5 k - 14$ using the AC method.

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Step 4.1

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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(k - 7) (k + 1)}{k (k + 1)} \cdot \frac{(k + 2) (k + 3)}{(k - 7) (k + 2)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $k - 7$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(k - 7) (k + 1)}{k (k + 1)} \cdot \frac{(k + 2) (k + 3)}{(k - 7) (k + 2)}$

Step 5.1.2

Rewrite the expression.

$\frac{k + 1}{k (k + 1)} \cdot \frac{(k + 2) (k + 3)}{k + 2}$

Step 5.2

Multiply $\frac{k + 1}{k (k + 1)}$ by $\frac{(k + 2) (k + 3)}{k + 2}$ .

$\frac{(k + 1) ((k + 2) (k + 3))}{k (k + 1) (k + 2)}$

Step 5.3

Cancel the common factor of $k + 1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(k + 1) ((k + 2) (k + 3))}{k (k + 1) (k + 2)}$

Step 5.3.2

Rewrite the expression.

$\frac{(k + 2) (k + 3)}{k (k + 2)}$

Step 5.4

Cancel the common factor of $k + 2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(k + 2) (k + 3)}{k (k + 2)}$

Step 5.4.2

Rewrite the expression.

$\frac{k + 3}{k}$