Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $a$ out of $- 3 a (4 a - 3)$ .

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

Step 1.1.2

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $4$ by $- 3$ .

$\frac{a (a^{2} + 3 a + 9 - 12 a - 3 \cdot - 3)}{(a - 3) (a^{2} + 3 a + 9)}$

Step 1.4

Multiply $- 3$ by $- 3$ .

$\frac{a (a^{2} + 3 a + 9 - 12 a + 9)}{(a - 3) (a^{2} + 3 a + 9)}$

Step 1.5

Subtract $12 a$ from $3 a$ .

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

Step 1.6

Add $9$ and $9$ .

$\frac{a (a^{2} - 9 a + 18)}{(a - 3) (a^{2} + 3 a + 9)}$

Step 1.7

Factor $a^{2} - 9 a + 18$ using the AC method.

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Step 1.7.1

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 $- 9$ .

$- 6, - 3$

Step 1.7.2

Write the factored form using these integers.

$\frac{a ((a - 6) (a - 3))}{(a - 3) (a^{2} + 3 a + 9)}$

$\frac{a (a - 6) (a - 3)}{(a - 3) (a^{2} + 3 a + 9)}$

Step 2

Cancel the common factor of $a - 3$ .

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Step 2.1

Cancel the common factor.

$\frac{a (a - 6) (a - 3)}{(a - 3) (a^{2} + 3 a + 9)}$

Step 2.2

Rewrite the expression.

$\frac{a (a - 6)}{a^{2} + 3 a + 9}$