Basic Math Examples

$\frac{a^{3} - 4 a}{a^{3} + 7 a^{2} + 10 a}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a$ out of $a^{3} - 4 a$ .

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Step 1.1.1

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

$\frac{a \cdot a^{2} - 4 a}{a^{3} + 7 a^{2} + 10 a}$

Step 1.1.2

Factor $a$ out of $- 4 a$ .

$\frac{a \cdot a^{2} + a \cdot - 4}{a^{3} + 7 a^{2} + 10 a}$

Step 1.1.3

Factor $a$ out of $a \cdot a^{2} + a \cdot - 4$ .

$\frac{a (a^{2} - 4)}{a^{3} + 7 a^{2} + 10 a}$

Step 1.2

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

$\frac{a (a^{2} - 2^{2})}{a^{3} + 7 a^{2} + 10 a}$

Step 1.3

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 $b = 2$ .

$\frac{a (a + 2) (a - 2)}{a^{3} + 7 a^{2} + 10 a}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $a$ out of $a^{3} + 7 a^{2} + 10 a$ .

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Step 2.1.1

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

$\frac{a (a + 2) (a - 2)}{a \cdot a^{2} + 7 a^{2} + 10 a}$

Step 2.1.2

Factor $a$ out of $7 a^{2}$ .

$\frac{a (a + 2) (a - 2)}{a \cdot a^{2} + a (7 a) + 10 a}$

Step 2.1.3

Factor $a$ out of $10 a$ .

$\frac{a (a + 2) (a - 2)}{a \cdot a^{2} + a (7 a) + a \cdot 10}$

Step 2.1.4

Factor $a$ out of $a \cdot a^{2} + a (7 a)$ .

$\frac{a (a + 2) (a - 2)}{a (a^{2} + 7 a) + a \cdot 10}$

Step 2.1.5

Factor $a$ out of $a (a^{2} + 7 a) + a \cdot 10$ .

$\frac{a (a + 2) (a - 2)}{a (a^{2} + 7 a + 10)}$

Step 2.2

Factor $a^{2} + 7 a + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Step 2.2.2

Write the factored form using these integers.

$\frac{a (a + 2) (a - 2)}{a ((a + 2) (a + 5))}$

$\frac{a (a + 2) (a - 2)}{a (a + 2) (a + 5)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $a$ .

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Step 3.1.1

Cancel the common factor.

$\frac{a (a + 2) (a - 2)}{a (a + 2) (a + 5)}$

Step 3.1.2

Rewrite the expression.

$\frac{(a + 2) (a - 2)}{(a + 2) (a + 5)}$

Step 3.2

Cancel the common factor of $a + 2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(a + 2) (a - 2)}{(a + 2) (a + 5)}$

Step 3.2.2

Rewrite the expression.

$\frac{a - 2}{a + 5}$