Algebra Examples

$\frac{2 g}{g^{2} - g - 6} - \frac{1}{g - 3}$

Step 1

Factor $g^{2} - g - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2

Write the factored form using these integers.

$\frac{2 g}{(g - 3) (g + 2)} - \frac{1}{g - 3}$

Step 2

To write $- \frac{1}{g - 3}$ as a fraction with a common denominator, multiply by $\frac{g + 2}{g + 2}$ .

$\frac{2 g}{(g - 3) (g + 2)} - \frac{1}{g - 3} \cdot \frac{g + 2}{g + 2}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{1}{g - 3}$ by $\frac{g + 2}{g + 2}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{2 g - g - 1 \cdot 2}{(g - 3) (g + 2)}$

Step 4.2

Multiply $- 1$ by $2$ .

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

Step 4.3

Subtract $g$ from $2 g$ .

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