- Identify the domain restriction \(x
eq 3\) by setting the denominators equal to zero.
- Factor the denominators and simplify the equation to \(x^2 = 3\).
- Solve for \(x\) to find \(x = \pm \sqrt{3}\).
- Verify that these values satisfy the original equation and are not equal to the restriction, confirming they are actual solutions: \(\boxed{x = \pm \sqrt{3}, \text{and they are actual solutions.}}\)
Explanation
- Analyzing the Equation and Domain Restrictions
The given rational equation is \(\frac{x^2}{2x-6} = \frac{9}{6x-18}\). First, we must identify the values of \(x\) that make the denominators zero, as these values are excluded from the domain of the equation. Setting the denominators to zero:
\(2x - 6 = 0 \implies 2(x - 3) = 0 \implies x = 3\)
\(6x - 18 = 0 \implies 6(x - 3) = 0 \implies x = 3\)
Thus, the equation is undefined at \(x = 3\). Any potential solution we find must be checked against this restriction to determine if it is an actual or extraneous solution.
- Simplifying and Solving the Rational Equation
We rewrite the equation by factoring the denominators:
\[\frac{x^2}{2(x-3)} = \frac{9}{6(x-3)}\]
To simplify the right side, we can reduce the fraction \(\frac{9}{6}\) to \(\frac{3}{2}\):
\[\frac{x^2}{2(x-3)} = \frac{3}{2(x-3)}\]
Assuming \(x
eq 3\), we can multiply both sides of the equation by \(2(x-3)\) to clear the denominators:
\[x^2 = 3\]
- Evaluating the Solutions
Solving the quadratic equation \(x^2 = 3\) gives two potential solutions:
\[x = \sqrt{3} \quad \text{or} \quad x = -\sqrt{3}\]
Since neither \(\sqrt{3}\) nor \(-\sqrt{3}\) is equal to 3, these values do not violate the domain restriction \(x
eq 3\). Therefore, both are valid, actual solutions to the original rational equation. Comparing these findings to the provided options, we conclude that \(x = \pm \sqrt{3}\) and they are actual solutions.
Examples
In engineering and physics, rational equations frequently arise when modeling physical systems like electrical circuits or fluid dynamics. Identifying domain restrictions (like avoiding division by zero) is crucial to ensure that the calculated values are physically meaningful and not extraneous artifacts of algebraic manipulation. Understanding this helps engineers prevent system failures by identifying and excluding non-realizable configurations in their designs.