Basic Math Examples

$\frac{a^{2} - 5 a - 14}{48 a^{2}} \cdot \frac{60 a^{3}}{a^{2} - 4 a - 21}$

Step 1

Factor $a^{2} - 5 a - 14$ 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(a - 7) (a + 2)}{48 a^{2}} \cdot \frac{60 a^{3}}{a^{2} - 4 a - 21}$

Step 2

Factor $a^{2} - 4 a - 21$ using the AC method.

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Step 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 $- 4$ .

$- 7, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 7) (a + 2)}{48 a^{2}} \cdot \frac{60 a^{3}}{(a - 7) (a + 3)}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{(a - 7) (a + 2) (60 a^{3})}{48 a^{2} ((a - 7) (a + 3))}$

Step 3.2

Cancel the common factor of $a - 7$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(a - 7) (a + 2) (60 a^{3})}{48 a^{2} ((a - 7) (a + 3))}$

Step 3.2.2

Rewrite the expression.

$\frac{(a + 2) (60 a^{3})}{48 a^{2} (a + 3)}$

Step 3.3

Cancel the common factor of $60$ and $48$ .

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Step 3.3.1

Factor $12$ out of $(a + 2) (60 a^{3})$ .

$\frac{12 ((a + 2) (5 a^{3}))}{48 a^{2} (a + 3)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $12$ out of $48 a^{2} (a + 3)$ .

$\frac{12 ((a + 2) (5 a^{3}))}{12 (4 a^{2} (a + 3))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{12 ((a + 2) (5 a^{3}))}{12 (4 a^{2} (a + 3))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{(a + 2) (5 a^{3})}{4 a^{2} (a + 3)}$

Step 3.4

Cancel the common factor of $a^{3}$ and $a^{2}$ .

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Step 3.4.1

Factor $a^{2}$ out of $(a + 2) (5 a^{3})$ .

$\frac{a^{2} ((a + 2) (5 a))}{4 a^{2} (a + 3)}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $a^{2}$ out of $4 a^{2} (a + 3)$ .

$\frac{a^{2} ((a + 2) (5 a))}{a^{2} (4 (a + 3))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{a^{2} ((a + 2) (5 a))}{a^{2} (4 (a + 3))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{(a + 2) (5 a)}{4 (a + 3)}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Move $5$ to the left of $a + 2$ .

$\frac{5 \cdot (a + 2) a}{4 (a + 3)}$

Step 3.5.2

Reorder factors in $\frac{5 (a + 2) a}{4 (a + 3)}$ .

$\frac{5 a (a + 2)}{4 (a + 3)}$