- Multiply both sides of the equation by \(k\) to clear the denominator, resulting in \(k^2 + 12 = 8k\).
- Move all terms to one side by subtracting \(8k\) from both sides to obtain the quadratic form.
- The resulting equation is \(k^2 - 8k + 12 = 0\).
- The correct option is \(\boxed{k^2 - 8k + 12 = 0}\).
Explanation
- Problem Analysis
The problem presents the equation \(k + \frac{12}{k} = 8\), where \(k
eq 0\). The goal is to transform this equation into a standard quadratic form \(ak^2 + bk + c = 0\) by clearing the denominator and grouping all terms on one side of the equal sign. This is a common technique used to solve rational equations.
- Clearing the Denominator
To eliminate the fraction \(\frac{12}{k}\), we multiply both sides of the equation by \(k\).
\[k \cdot \left(k + \frac{12}{k}\right) = k \cdot 8\]
Distributing \(k\) on the left side gives:
\[k \cdot k + k \cdot \frac{12}{k} = 8k\]
\[k^2 + 12 = 8k\]
- Rearranging and Selecting the Result
Now, move all terms to one side to set the equation to zero by subtracting \(8k\) from both sides:
\[k^2 - 8k + 12 = 0\]
Comparing this result with the given options, we identify that the correct equation is \(k^2 - 8k + 12 = 0\).
Examples
In physics and engineering, rational equations often appear when dealing with inverse relationships, such as resistance in parallel circuits or the lens equation in optics. For instance, when solving for the focal length or distance of an object, you often end up with an equation involving reciprocals. Clearing denominators by multiplying by a common variable helps convert these into familiar quadratic equations that are easily solvable using the quadratic formula or factoring. This process is essential for simplifying complex physical models into manageable calculations.