Find Where Undefined/Discontinuous 1-5/(x-2)=-20/(x^2-4)

$1 - \frac{5}{x - 2} = - \frac{20}{x^{2} - 4}$

Step 1

Add $\frac{20}{x^{2} - 4}$ to both sides of the equation.

$1 - \frac{5}{x - 2} + \frac{20}{x^{2} - 4} = 0$

Step 2

Simplify $1 - \frac{5}{x - 2} + \frac{20}{x^{2} - 4}$ .

Simplify the denominator.

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

$1 - \frac{5}{x - 2} + \frac{20}{x^{2} - 2^{2}} = 0$

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

$1 - \frac{5}{x - 2} + \frac{20}{(x + 2) (x - 2)} = 0$

Write $1$ as a fraction with a common denominator.

$\frac{x - 2}{x - 2} - \frac{5}{x - 2} + \frac{20}{(x + 2) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{x - 2 - 5}{x - 2} + \frac{20}{(x + 2) (x - 2)} = 0$

Subtract $5$ from $- 2$ .

$\frac{x - 7}{x - 2} + \frac{20}{(x + 2) (x - 2)} = 0$

To write $\frac{x - 7}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x - 7}{x - 2} \cdot \frac{x + 2}{x + 2} + \frac{20}{(x + 2) (x - 2)} = 0$

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

Multiply $\frac{x - 7}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{(x - 7) (x + 2)}{(x - 2) (x + 2)} + \frac{20}{(x + 2) (x - 2)} = 0$

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

$\frac{(x - 7) (x + 2)}{(x + 2) (x - 2)} + \frac{20}{(x + 2) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(x - 7) (x + 2) + 20}{(x + 2) (x - 2)} = 0$

Simplify the numerator.

Expand $(x - 7) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 2) - 7 (x + 2) + 20}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 7 (x + 2) + 20}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 7 x - 7 \cdot 2 + 20}{(x + 2) (x - 2)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - 7 x - 7 \cdot 2 + 20}{(x + 2) (x - 2)} = 0$

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x - 7 x - 7 \cdot 2 + 20}{(x + 2) (x - 2)} = 0$

Multiply $- 7$ by $2$ .

$\frac{x^{2} + 2 x - 7 x - 14 + 20}{(x + 2) (x - 2)} = 0$

Subtract $7 x$ from $2 x$ .

$\frac{x^{2} - 5 x - 14 + 20}{(x + 2) (x - 2)} = 0$

Add $- 14$ and $20$ .

$\frac{x^{2} - 5 x + 6}{(x + 2) (x - 2)} = 0$

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 3}{x + 2} = 0$

Step 3

Set the denominator in $\frac{x - 3}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Find Where Undefined/Discontinuous 1-5/(x-2)=-20/(x^2-4)

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