Basic Math Examples

$\frac{a - 2}{a + 3} + \frac{10 a}{a^{2} - 9}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.2

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 $b = 3$ .

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

Step 2

To write $\frac{a - 2}{a + 3}$ as a fraction with a common denominator, multiply by $\frac{a - 3}{a - 3}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{a - 2}{a + 3}$ by $\frac{a - 3}{a - 3}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Expand $(a - 2) (a - 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $a$ by $a$ .

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

Step 4.2.1.2

Move $- 3$ to the left of $a$ .

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

Step 4.2.1.3

Multiply $- 2$ by $- 3$ .

$\frac{a^{2} - 3 a - 2 a + 6 + 10 a}{(a + 3) (a - 3)}$

Step 4.2.2

Subtract $2 a$ from $- 3 a$ .

$\frac{a^{2} - 5 a + 6 + 10 a}{(a + 3) (a - 3)}$

Step 4.3

Add $- 5 a$ and $10 a$ .

$\frac{a^{2} + 5 a + 6}{(a + 3) (a - 3)}$

Step 4.4

Factor $a^{2} + 5 a + 6$ using the AC method.

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Step 4.4.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 4.4.2

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $a + 3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{a + 2}{a - 3}$