- Set the denominator \(2x^2 - 7x - 4\) to zero to find the excluded values.
- Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with coefficients \(a=2, b=-7, c=-4\).
- Simplify to obtain the roots \(x = -1/2\) and \(x = 4\).
- State the excluded values as \(x
eq -1/2, 4\), with the smallest excluded value being \(-1/2\).
- The final answer is \(\boxed{x
eq -\frac{1}{2}, 4}\).
Explanation
- Identifying the Condition for Excluded Values
To find the excluded values of the rational expression \(\frac{5 x^2 + 5 x + 2}{2 x^2 - 7 x - 4}\), we must determine the values of \(x\) for which the denominator is equal to zero, as division by zero is undefined. We set the denominator to zero:
\[ 2x^2 - 7x - 4 = 0 \]
- Solving the Quadratic Equation
We solve the quadratic equation \(2x^2 - 7x - 4 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -7\), and \(c = -4\). Substituting these values into the formula gives:
\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(-4)}}{2(2)} \]
\[ x = \frac{7 \pm \sqrt{49 + 32}}{4} \]
\[ x = \frac{7 \pm \sqrt{81}}{4} \]
\[ x = \frac{7 \pm 9}{4} \]
This yields two possible values for \(x\):
\[ x_1 = \frac{7 - 9}{4} = \frac{-2}{4} = -\frac{1}{2} \]
\[ x_2 = \frac{7 + 9}{4} = \frac{16}{4} = 4 \]
- Conclusion and Final Answer Formatting
The excluded values for the expression are \(x = -1/2\) and \(x = 4\). The problem asks to list these values, starting with the smallest. Since \(-1/2 < 4\), the excluded values in increasing order are \(-1/2\) and \(4\). Thus, the expression is defined for all real numbers except \(x = -1/2\) and \(x = 4\). The negative excluded value is expressed as \(x
eq - \frac{1}{2}\).
Examples
In real-world engineering or data analysis, rational functions are frequently used to model physical systems, such as resistance in a circuit or concentrations in a chemical reaction. Excluded values represent physical limitations—such as points where a system might become unstable, overload, or reach a point of infinite pressure or flow. Identifying these "undefined" points ensures that engineers or scientists can safely operate within the valid domain, preventing errors or failures in practical applications.