Basic Math Examples

$\frac{k^{3} - k^{2} - 42 k}{2 k^{2} - 20 k + 42}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $k$ out of $k^{3} - k^{2} - 42 k$ .

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Step 1.1.1

Factor $k$ out of $k^{3}$ .

$\frac{k \cdot k^{2} - k^{2} - 42 k}{2 k^{2} - 20 k + 42}$

Step 1.1.2

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

$\frac{k \cdot k^{2} + k (- k) - 42 k}{2 k^{2} - 20 k + 42}$

Step 1.1.3

Factor $k$ out of $- 42 k$ .

$\frac{k \cdot k^{2} + k (- k) + k \cdot - 42}{2 k^{2} - 20 k + 42}$

Step 1.1.4

Factor $k$ out of $k \cdot k^{2} + k (- k)$ .

$\frac{k (k^{2} - k) + k \cdot - 42}{2 k^{2} - 20 k + 42}$

Step 1.1.5

Factor $k$ out of $k (k^{2} - k) + k \cdot - 42$ .

$\frac{k (k^{2} - k - 42)}{2 k^{2} - 20 k + 42}$

Step 1.2

Factor $k^{2} - k - 42$ using the AC method.

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Step 1.2.1

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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 1.2.2

Write the factored form using these integers.

$\frac{k ((k - 7) (k + 6))}{2 k^{2} - 20 k + 42}$

$\frac{k (k - 7) (k + 6)}{2 k^{2} - 20 k + 42}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $2 k^{2} - 20 k + 42$ .

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Step 2.1.1

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

$\frac{k (k - 7) (k + 6)}{2 (k^{2}) - 20 k + 42}$

Step 2.1.2

Factor $2$ out of $- 20 k$ .

$\frac{k (k - 7) (k + 6)}{2 (k^{2}) + 2 (- 10 k) + 42}$

Step 2.1.3

Factor $2$ out of $42$ .

$\frac{k (k - 7) (k + 6)}{2 k^{2} + 2 (- 10 k) + 2 \cdot 21}$

Step 2.1.4

Factor $2$ out of $2 k^{2} + 2 (- 10 k)$ .

$\frac{k (k - 7) (k + 6)}{2 (k^{2} - 10 k) + 2 \cdot 21}$

Step 2.1.5

Factor $2$ out of $2 (k^{2} - 10 k) + 2 \cdot 21$ .

$\frac{k (k - 7) (k + 6)}{2 (k^{2} - 10 k + 21)}$

Step 2.2

Factor $k^{2} - 10 k + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{k (k - 7) (k + 6)}{2 ((k - 7) (k - 3))}$

$\frac{k (k - 7) (k + 6)}{2 (k - 7) (k - 3)}$

Step 3

Cancel the common factor of $k - 7$ .

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Step 3.1

Cancel the common factor.

$\frac{k (k - 7) (k + 6)}{2 (k - 7) (k - 3)}$

Step 3.2

Rewrite the expression.

$\frac{k (k + 6)}{2 (k - 3)}$