Algebra Examples

$\frac{3}{2 g + 8} = \frac{g + 2}{g^{2} - 16}$

Step 1

Subtract $\frac{g + 2}{g^{2} - 16}$ from both sides of the equation.

$\frac{3}{2 g + 8} - \frac{g + 2}{g^{2} - 16} = 0$

Step 2

Simplify $\frac{3}{2 g + 8} - \frac{g + 2}{g^{2} - 16}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $2$ out of $2 g + 8$ .

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Step 2.1.1.1

Factor $2$ out of $2 g$ .

$\frac{3}{2 (g) + 8} - \frac{g + 2}{g^{2} - 16} = 0$

Step 2.1.1.2

Factor $2$ out of $8$ .

$\frac{3}{2 g + 2 \cdot 4} - \frac{g + 2}{g^{2} - 16} = 0$

Step 2.1.1.3

Factor $2$ out of $2 g + 2 \cdot 4$ .

$\frac{3}{2 (g + 4)} - \frac{g + 2}{g^{2} - 16} = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{3}{2 (g + 4)} - \frac{g + 2}{g^{2} - 4^{2}} = 0$

Step 2.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = g$ and $b = 4$ .

$\frac{3}{2 (g + 4)} - \frac{g + 2}{(g + 4) (g - 4)} = 0$

Step 2.2

To write $\frac{3}{2 (g + 4)}$ as a fraction with a common denominator, multiply by $\frac{g - 4}{g - 4}$ .

$\frac{3}{2 (g + 4)} \cdot \frac{g - 4}{g - 4} - \frac{g + 2}{(g + 4) (g - 4)} = 0$

Step 2.3

To write $- \frac{g + 2}{(g + 4) (g - 4)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2 (g + 4)} \cdot \frac{g - 4}{g - 4} - \frac{g + 2}{(g + 4) (g - 4)} \cdot \frac{2}{2} = 0$

Step 2.4

Write each expression with a common denominator of $2 (g + 4) (g - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{3}{2 (g + 4)}$ by $\frac{g - 4}{g - 4}$ .

$\frac{3 (g - 4)}{2 (g + 4) (g - 4)} - \frac{g + 2}{(g + 4) (g - 4)} \cdot \frac{2}{2} = 0$

Step 2.4.2

Multiply $\frac{g + 2}{(g + 4) (g - 4)}$ by $\frac{2}{2}$ .

$\frac{3 (g - 4)}{2 (g + 4) (g - 4)} - \frac{(g + 2) \cdot 2}{(g + 4) (g - 4) \cdot 2} = 0$

Step 2.4.3

Reorder the factors of $(g + 4) (g - 4) \cdot 2$ .

$\frac{3 (g - 4)}{2 (g + 4) (g - 4)} - \frac{(g + 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 (g - 4) - (g + 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{3 g + 3 \cdot - 4 - (g + 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.2

Multiply $3$ by $- 4$ .

$\frac{3 g - 12 - (g + 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.3

Apply the distributive property.

$\frac{3 g - 12 + (- g - 1 \cdot 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.4

Multiply $- 1$ by $2$ .

$\frac{3 g - 12 + (- g - 2) \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.5

Apply the distributive property.

$\frac{3 g - 12 - g \cdot 2 - 2 \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.6

Multiply $2$ by $- 1$ .

$\frac{3 g - 12 - 2 g - 2 \cdot 2}{2 (g + 4) (g - 4)} = 0$

Step 2.6.7

Multiply $- 2$ by $2$ .

$\frac{3 g - 12 - 2 g - 4}{2 (g + 4) (g - 4)} = 0$

Step 2.6.8

Subtract $2 g$ from $3 g$ .

$\frac{g - 12 - 4}{2 (g + 4) (g - 4)} = 0$

Step 2.6.9

Subtract $4$ from $- 12$ .

$\frac{g - 16}{2 (g + 4) (g - 4)} = 0$

Step 3

Set the denominator in $\frac{g - 16}{2 (g + 4) (g - 4)}$ equal to $0$ to find where the expression is undefined.

$2 (g + 4) (g - 4) = 0$

Step 4

Solve for $g$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$g + 4 = 0$

$g - 4 = 0$

Step 4.2

Set $g + 4$ equal to $0$ and solve for $g$ .

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Step 4.2.1

Set $g + 4$ equal to $0$ .

$g + 4 = 0$

Step 4.2.2

Subtract $4$ from both sides of the equation.

$g = - 4$

Step 4.3

Set $g - 4$ equal to $0$ and solve for $g$ .

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Step 4.3.1

Set $g - 4$ equal to $0$ .

$g - 4 = 0$

Step 4.3.2

Add $4$ to both sides of the equation.

$g = 4$

Step 4.4

The final solution is all the values that make $2 (g + 4) (g - 4) = 0$ true.

$g = - 4, 4$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$g = - 4, g = 4$

Step 6