Answered

Find the excluded value(s) of the rational expression $\frac{x^2-x-6}{3 x^2-48}$.



Set the denominator equal to 0:

$3 x^2-48=0$



Solve for $x$:

$\begin{aligned}

3 x^2 & =48 \\

x^2 & =16

\end{aligned}$



What value(s) for $x$ will make the rational expression undefined?


A$\pm 16$
B$\pm 4$
C-2, 3
D0

Verified Answer :

  • A rational expression is undefined when the denominator is zero.
  • Setting the denominator \(3x^2 - 48 = 0\) leads to \(x^2 = 16\).
  • Solving for \(x\) gives \(x = 4\) and \(x = -4\).
  • The excluded values are \(\boxed{\pm 4}\).

Explanation

  1. Understanding the Problem

A rational expression of the form \(\frac{P(x)}{Q(x)}\) is undefined when its denominator \(Q(x)\) is equal to zero. In the given expression \(\frac{x^2-x-6}{3x^2-48}\), the denominator is \(3x^2-48\). To find the values for which the expression is undefined, we must solve for \(x\) in the equation \(3x^2-48=0\).

  1. Solving for Excluded Values

We start by setting the denominator equal to zero:
\[3x^2 - 48 = 0\]
Next, we solve this quadratic equation for \(x\). First, we isolate the \(x^2\) term by adding 48 to both sides:
\[3x^2 = 48\]
Now, divide both sides by 3:
\[x^2 = \frac{48}{3} = 16\]
Taking the square root of both sides gives us two possible values for \(x\):
\[x = \pm \sqrt{16} = \pm 4\]
Thus, the values that make the expression undefined are \(x = 4\) and \(x = -4\).

  1. Conclusion and Final Answer

Based on our calculation, the denominator becomes zero when \(x=4\) or \(x=-4\). Therefore, the rational expression \(\frac{x^2-x-6}{3x^2-48}\) is undefined at these specific points. Comparing this to the given choices, the correct set of excluded values is \(\pm 4\).

Examples

Rational expressions appear frequently in physics and engineering when modeling rates or proportions. For example, if you are modeling the speed of an object where the time taken depends on the distance divided by speed, say \(v = \frac{d}{t}\), if the time variable were in the denominator as an expression, identifying the values for which that expression equals zero is critical. It tells the engineer where the physical model breaks down or becomes undefined, such as a point of infinite speed or a division by zero error in a computer simulation.

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