Algebra Examples

$\frac{4 a + 1}{a^{2} - 4} - \frac{2 a + 7}{a^{2} - 4}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{4 a + 1 - (2 a + 7)}{a^{2} - 4}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{4 a + 1 - (2 a) - 1 \cdot 7}{a^{2} - 4}$

Step 1.2.2

Multiply $2$ by $- 1$ .

$\frac{4 a + 1 - 2 a - 1 \cdot 7}{a^{2} - 4}$

Step 1.2.3

Multiply $- 1$ by $7$ .

$\frac{4 a + 1 - 2 a - 7}{a^{2} - 4}$

Step 1.3

Simplify terms.

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Step 1.3.1

Subtract $2 a$ from $4 a$ .

$\frac{2 a + 1 - 7}{a^{2} - 4}$

Step 1.3.2

Subtract $7$ from $1$ .

$\frac{2 a - 6}{a^{2} - 4}$

Step 1.3.3

Factor $2$ out of $2 a - 6$ .

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Step 1.3.3.1

Factor $2$ out of $2 a$ .

$\frac{2 (a) - 6}{a^{2} - 4}$

Step 1.3.3.2

Factor $2$ out of $- 6$ .

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

Step 1.3.3.3

Factor $2$ out of $2 a + 2 \cdot - 3$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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 $b = 2$ .

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