Basic Math Examples

$\frac{a^{2} - 7 a - 18}{4 a^{3}} \div \frac{a^{2} - 4 a - 45}{2 a^{3} - 4 a^{2}} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 1

Factor $a^{2} - 7 a - 18$ 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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{a^{2} - 4 a - 45}{2 a^{3} - 4 a^{2}} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 2

Factor $a^{2} - 4 a - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{3} - 4 a^{2}} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 3

Factor $2 a^{2}$ out of $2 a^{3} - 4 a^{2}$ .

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Step 3.1

Factor $2 a^{2}$ out of $2 a^{3}$ .

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a) - 4 a^{2}} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 3.2

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

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a) + 2 a^{2} (- 2)} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 3.3

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

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} \cdot \frac{a^{2} + 3 a - 10}{a^{2} - 4 a + 4}$

Step 4

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

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Step 4.1

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 $3$ .

$- 2, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} \cdot \frac{(a - 2) (a + 5)}{a^{2} - 4 a + 4}$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} \cdot \frac{(a - 2) (a + 5)}{a^{2} - 4 a + 2^{2}}$

Step 5.2

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

$4 a = 2 \cdot a \cdot 2$

Step 5.3

Rewrite the polynomial.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} \cdot \frac{(a - 2) (a + 5)}{a^{2} - 2 \cdot a \cdot 2 + 2^{2}}$

Step 5.4

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

$\frac{(a - 9) (a + 2)}{4 a^{3}} \div \frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} \cdot \frac{(a - 2) (a + 5)}{{(a - 2)}^{2}}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{\frac{(a - 9) (a + 2)}{4 a^{3}} ((a - 2) (a + 5))}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} {(a - 2)}^{2}}$

Step 6.2

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

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Step 6.2.1

Factor $a - 2$ out of $\frac{(a - 9) (a + 2)}{4 a^{3}} ((a - 2) (a + 5))$ .

$\frac{(a - 2) (\frac{(a - 9) (a + 2)}{4 a^{3}} (a + 5))}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} {(a - 2)}^{2}}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $a - 2$ out of $\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} {(a - 2)}^{2}$ .

$\frac{(a - 2) (\frac{(a - 9) (a + 2)}{4 a^{3}} (a + 5))}{(a - 2) (\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2))}$

Step 6.2.2.2

Cancel the common factor.

$\frac{(a - 2) (\frac{(a - 9) (a + 2)}{4 a^{3}} (a + 5))}{(a - 2) (\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2))}$

Step 6.2.2.3

Rewrite the expression.

$\frac{\frac{(a - 9) (a + 2)}{4 a^{3}} (a + 5)}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2)}$

Step 6.3

Factor $\frac{(a - 9) (a + 2)}{4 a^{3}}$ out of $\frac{\frac{(a - 9) (a + 2)}{4 a^{3}} (a + 5)}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2)}$ .

$\frac{(a - 9) (a + 2)}{4 a^{3}} \cdot \frac{a + 5}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2)}$

Step 7

Factor $\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)}$ out of $\frac{a + 5}{\frac{(a - 9) (a + 5)}{2 a^{2} (a - 2)} (a - 2)}$ .

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{2 a^{2} (a - 2)}{(a - 9) (a + 5)} \cdot \frac{a + 5}{a - 2})$

Step 8

Cancel the common factor of $a - 2$ .

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Step 8.1

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

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{(a - 2) (2 a^{2})}{(a - 9) (a + 5)} \cdot \frac{a + 5}{a - 2})$

Step 8.2

Cancel the common factor.

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{(a - 2) (2 a^{2})}{(a - 9) (a + 5)} \cdot \frac{a + 5}{a - 2})$

Step 8.3

Rewrite the expression.

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{2 a^{2}}{(a - 9) (a + 5)} (a + 5))$

Step 9

Cancel the common factor of $a + 5$ .

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Step 9.1

Factor $a + 5$ out of $(a - 9) (a + 5)$ .

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{2 a^{2}}{(a + 5) (a - 9)} (a + 5))$

Step 9.2

Cancel the common factor.

$\frac{(a - 9) (a + 2)}{4 a^{3}} (\frac{2 a^{2}}{(a + 5) (a - 9)} (a + 5))$

Step 9.3

Rewrite the expression.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \cdot \frac{2 a^{2}}{a - 9}$

Step 10

Cancel the common factor of $a - 9$ .

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Step 10.1

Cancel the common factor.

$\frac{(a - 9) (a + 2)}{4 a^{3}} \cdot \frac{2 a^{2}}{a - 9}$

Step 10.2

Rewrite the expression.

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

Step 11

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

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Step 11.1

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

$\frac{a + 2}{2 a^{2} (2 a)} (2 a^{2})$

Step 11.2

Cancel the common factor.

$\frac{a + 2}{2 a^{2} (2 a)} (2 a^{2})$

Step 11.3

Rewrite the expression.

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