Find Where Undefined/Discontinuous 3/(a+2)+2/a=(4a-4)/(a^2-4)

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

Step 1

Subtract $\frac{4 a - 4}{a^{2} - 4}$ from both sides of the equation.

$\frac{3}{a + 2} + \frac{2}{a} - \frac{4 a - 4}{a^{2} - 4} = 0$

Step 2

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

Simplify each term.

Factor $4$ out of $4 a - 4$ .

Factor $4$ out of $4 a$ .

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

Factor $4$ out of $- 4$ .

$\frac{3}{a + 2} + \frac{2}{a} - \frac{4 (a) + 4 (- 1)}{a^{2} - 4} = 0$

Factor $4$ out of $4 (a) + 4 (- 1)$ .

$\frac{3}{a + 2} + \frac{2}{a} - \frac{4 (a - 1)}{a^{2} - 4} = 0$

Simplify the denominator.

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

$\frac{3}{a + 2} + \frac{2}{a} - \frac{4 (a - 1)}{a^{2} - 2^{2}} = 0$

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 $b = 2$ .

$\frac{3}{a + 2} + \frac{2}{a} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

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

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

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

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

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

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

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

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

$\frac{3 a}{(a + 2) a} + \frac{2 (a + 2)}{a (a + 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

Reorder the factors of $(a + 2) a$ .

$\frac{3 a}{a (a + 2)} + \frac{2 (a + 2)}{a (a + 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{3 a + 2 (a + 2)}{a (a + 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $2$ .

$\frac{3 a + 2 a + 4}{a (a + 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

Add $3 a$ and $2 a$ .

$\frac{5 a + 4}{a (a + 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

To write $\frac{5 a + 4}{a (a + 2)}$ as a fraction with a common denominator, multiply by $\frac{a - 2}{a - 2}$ .

$\frac{5 a + 4}{a (a + 2)} \cdot \frac{a - 2}{a - 2} - \frac{4 (a - 1)}{(a + 2) (a - 2)} = 0$

To write $- \frac{4 (a - 1)}{(a + 2) (a - 2)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{5 a + 4}{a (a + 2)} \cdot \frac{a - 2}{a - 2} - \frac{4 (a - 1)}{(a + 2) (a - 2)} \cdot \frac{a}{a} = 0$

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

Multiply $\frac{5 a + 4}{a (a + 2)}$ by $\frac{a - 2}{a - 2}$ .

$\frac{(5 a + 4) (a - 2)}{a (a + 2) (a - 2)} - \frac{4 (a - 1)}{(a + 2) (a - 2)} \cdot \frac{a}{a} = 0$

Multiply $\frac{4 (a - 1)}{(a + 2) (a - 2)}$ by $\frac{a}{a}$ .

$\frac{(5 a + 4) (a - 2)}{a (a + 2) (a - 2)} - \frac{4 (a - 1) a}{(a + 2) (a - 2) a} = 0$

Reorder the factors of $(a + 2) (a - 2) a$ .

$\frac{(5 a + 4) (a - 2)}{a (a + 2) (a - 2)} - \frac{4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(5 a + 4) (a - 2) - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Simplify the numerator.

Expand $(5 a + 4) (a - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 a (a - 2) + 4 (a - 2) - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Apply the distributive property.

$\frac{5 a \cdot a + 5 a \cdot - 2 + 4 (a - 2) - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Apply the distributive property.

$\frac{5 a \cdot a + 5 a \cdot - 2 + 4 a + 4 \cdot - 2 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{5 (a \cdot a) + 5 a \cdot - 2 + 4 a + 4 \cdot - 2 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Multiply $a$ by $a$ .

$\frac{5 a^{2} + 5 a \cdot - 2 + 4 a + 4 \cdot - 2 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Multiply $- 2$ by $5$ .

$\frac{5 a^{2} - 10 a + 4 a + 4 \cdot - 2 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Multiply $4$ by $- 2$ .

$\frac{5 a^{2} - 10 a + 4 a - 8 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Add $- 10 a$ and $4 a$ .

$\frac{5 a^{2} - 6 a - 8 - 4 (a - 1) a}{a (a + 2) (a - 2)} = 0$

Apply the distributive property.

$\frac{5 a^{2} - 6 a - 8 + (- 4 a - 4 \cdot - 1) a}{a (a + 2) (a - 2)} = 0$

Multiply $- 4$ by $- 1$ .

$\frac{5 a^{2} - 6 a - 8 + (- 4 a + 4) a}{a (a + 2) (a - 2)} = 0$

Apply the distributive property.

$\frac{5 a^{2} - 6 a - 8 - 4 a \cdot a + 4 a}{a (a + 2) (a - 2)} = 0$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{5 a^{2} - 6 a - 8 - 4 (a \cdot a) + 4 a}{a (a + 2) (a - 2)} = 0$

Multiply $a$ by $a$ .

$\frac{5 a^{2} - 6 a - 8 - 4 a^{2} + 4 a}{a (a + 2) (a - 2)} = 0$

Subtract $4 a^{2}$ from $5 a^{2}$ .

$\frac{a^{2} - 6 a - 8 + 4 a}{a (a + 2) (a - 2)} = 0$

Add $- 6 a$ and $4 a$ .

$\frac{a^{2} - 2 a - 8}{a (a + 2) (a - 2)} = 0$

Factor $a^{2} - 2 a - 8$ using the AC method.

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

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

Cancel the common factor of $a + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{a - 4}{a (a - 2)} = 0$

Step 3

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

$a (a - 2) = 0$

Step 4

Solve for $a$ .

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

$a = 0$

$a - 2 = 0$

Set $a$ equal to $0$ .

$a = 0$

Set $a - 2$ equal to $0$ and solve for $a$ .

Set $a - 2$ equal to $0$ .

$a - 2 = 0$

Add $2$ to both sides of the equation.

$a = 2$

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

$a = 0, 2$

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 $0$ .

$a = 0, a = 2$

Step 6

Find Where Undefined/Discontinuous 3/(a+2)+2/a=(4a-4)/(a^2-4)

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