Basic Math Examples

$\frac{a^{2} - 14 a + 49}{a^{2} - a - 42} \div \frac{a^{2} - 49}{7}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - 14 a + 49}{a^{2} - a - 42} \cdot \frac{7}{a^{2} - 49}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{a^{2} - 14 a + 7^{2}}{a^{2} - a - 42} \cdot \frac{7}{a^{2} - 49}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 a = 2 \cdot a \cdot 7$

Step 2.3

Rewrite the polynomial.

$\frac{a^{2} - 2 \cdot a \cdot 7 + 7^{2}}{a^{2} - a - 42} \cdot \frac{7}{a^{2} - 49}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 7$ .

$\frac{{(a - 7)}^{2}}{a^{2} - a - 42} \cdot \frac{7}{a^{2} - 49}$

Step 3

Factor $a^{2} - a - 42$ 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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 3.2

Write the factored form using these integers.

$\frac{{(a - 7)}^{2}}{(a - 7) (a + 6)} \cdot \frac{7}{a^{2} - 49}$

Step 4

Cancel the common factor of ${(a - 7)}^{2}$ and $a - 7$ .

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Step 4.1

Factor $a - 7$ out of ${(a - 7)}^{2}$ .

$\frac{(a - 7) (a - 7)}{(a - 7) (a + 6)} \cdot \frac{7}{a^{2} - 49}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Cancel the common factor.

$\frac{(a - 7) (a - 7)}{(a - 7) (a + 6)} \cdot \frac{7}{a^{2} - 49}$

Step 4.2.2

Rewrite the expression.

$\frac{a - 7}{a + 6} \cdot \frac{7}{a^{2} - 49}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $49$ as $7^{2}$ .

$\frac{a - 7}{a + 6} \cdot \frac{7}{a^{2} - 7^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 7$ .

$\frac{a - 7}{a + 6} \cdot \frac{7}{(a + 7) (a - 7)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $a - 7$ .

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Step 6.1.1

Factor $a - 7$ out of $(a + 7) (a - 7)$ .

$\frac{a - 7}{a + 6} \cdot \frac{7}{(a - 7) (a + 7)}$

Step 6.1.2

Cancel the common factor.

$\frac{a - 7}{a + 6} \cdot \frac{7}{(a - 7) (a + 7)}$

Step 6.1.3

Rewrite the expression.

$\frac{1}{a + 6} \cdot \frac{7}{a + 7}$

Step 6.2

Multiply $\frac{1}{a + 6}$ by $\frac{7}{a + 7}$ .

$\frac{7}{(a + 6) (a + 7)}$