Questions: Find the restricted values of (x) for the following rational expression. [ frac-2 xx^2-5 x ]

$Find the restricted values of (x) for the following rational expression. [ frac-2 xx^2-5 x ]$

Transcript text: Find the restricted values of $x$ for the following rational expression. \[ \frac{-2 x}{x^{2}-5 x} \]

Solution

Solution Steps

To find the restricted values of $ x $ for the given rational expression, we need to determine the values of $ x $ that make the denominator equal to zero. These values are the ones that will make the expression undefined.

Step 1: Identify the Denominator

The given rational expression is

\[ \frac{-2x}{x^2 - 5x} \]

To find the restricted values of $ x $, we need to analyze the denominator $ x^2 - 5x $.

Step 2: Set the Denominator to Zero

We set the denominator equal to zero to find the values of $ x $ that make the expression undefined:

\[ x^2 - 5x = 0 \]

Step 3: Factor the Denominator

Factoring the equation gives us:

\[ x(x - 5) = 0 \]

Step 4: Solve for $ x $

Setting each factor to zero, we find:

\[ x = 0 \quad \text{or} \quad x - 5 = 0 \implies x = 5 \]

Final Answer

The restricted values of $ x $ are

\[ \boxed{x = 0} \quad \text{and} \quad \boxed{x = 5} \]

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