Question
State the excluded values of the given rational expression: \( \frac{x^{2}-5 x+3}{x^{2}-7 x+12} \)
Ask by Klein Munoz. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The excluded values are 3 and 4.
Solution
To determine the **excluded values** of the rational expression:
\[
\frac{x^{2} - 5x + 3}{x^{2} - 7x + 12}
\]
we need to identify the values of \( x \) that make the **denominator zero**. These values are excluded because division by zero is undefined.
**Step 1: Set the denominator equal to zero and solve for \( x \):**
\[
x^{2} - 7x + 12 = 0
\]
**Step 2: Factor the quadratic equation:**
\[
x^{2} - 7x + 12 = (x - 3)(x - 4) = 0
\]
**Step 3: Solve for \( x \):**
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
**Conclusion:**
The **excluded values** of the rational expression are:
\[
\boxed{3 \text{ and } 4}
\]
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Simplify this solution The Deep Dive
To find the excluded values of the rational expression, we need to determine the values of \(x\) that make the denominator zero. The denominator is \(x^{2} - 7x + 12\). Factoring this polynomial gives us \((x - 3)(x - 4)\), which means the values that make the denominator zero are \(x = 3\) and \(x = 4\). Therefore, the excluded values are \(x = 3\) and \(x = 4\). Additionally, rational expressions cannot take on values that cause division by zero. This is why we have to set the denominator equal to zero and solve for \(x\). It’s a good reminder that excluded values often arise from simple quadratic factors!
