Simplify ((k+5)/(k^2+3k-10))÷((7k+14)/(4-k^2))

$\frac{k + 5}{k^{2} + 3 k - 10} \div \frac{7 k + 14}{4 - k^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{k + 5}{k^{2} + 3 k - 10} \cdot \frac{4 - k^{2}}{7 k + 14}$

Step 2

Factor $k^{2} + 3 k - 10$ using the AC method.

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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{k + 5}{(k - 2) (k + 5)} \cdot \frac{4 - k^{2}}{7 k + 14}$

Step 3

Cancel the common factor of $k + 5$ .

Cancel the common factor.

$\frac{k + 5}{(k - 2) (k + 5)} \cdot \frac{4 - k^{2}}{7 k + 14}$

Rewrite the expression.

$\frac{1}{k - 2} \cdot \frac{4 - k^{2}}{7 k + 14}$

Step 4

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{1}{k - 2} \cdot \frac{2^{2} - k^{2}}{7 k + 14}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = k$ .

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

Step 5

Simplify terms.

Factor $7$ out of $7 k + 14$ .

Factor $7$ out of $7 k$ .

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

Factor $7$ out of $14$ .

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

Factor $7$ out of $7 k + 7 \cdot 2$ .

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

Cancel the common factor of $2 + k$ and $k + 2$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{k - 2} \cdot \frac{2 - k}{7}$

Multiply $\frac{1}{k - 2}$ by $\frac{2 - k}{7}$ .

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

Cancel the common factor of $2 - k$ and $k - 2$ .

Rewrite $2$ as $- 1 (- 2)$ .

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

Factor $- 1$ out of $- k$ .

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

Factor $- 1$ out of $- 1 (- 2) - (k)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1}{7}$

Move the negative in front of the fraction.

$- \frac{1}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{7}$

Decimal Form:

$- 0.\overline{142857}$

Simplify ((k+5)/(k^2+3k-10))÷((7k+14)/(4-k^2))

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