Find the real solution of the following radical equation:
$\sqrt{3x} - 6 = 6$
To find the real solution of the equation \(\sqrt{3x} - 6 = 6\), we first analyze the structure of the equation. It is a radical equation involving a square root. For the expression \(\sqrt{3x}\) to be a real number, we must satisfy the condition \(3x \geq 0\), which simplifies to \(x \geq 0\). This ensures that the expression inside the square root is non-negative, allowing for real solutions.
Our first step is to isolate the radical term \(\sqrt{3x}\) on one side of the equation. Given the equation \(\sqrt{3x} - 6 = 6\), we add 6 to both sides:
\[\sqrt{3x} = 6 + 6\]
\[\sqrt{3x} = 12\]
Now that the radical is isolated, we square both sides to eliminate the square root. Squaring both sides gives:
\[(\sqrt{3x})^2 = 12^2\]
\[3x = 144\]
Now, we solve for \(x\) by dividing both sides by 3:
\[x = \frac{144}{3}\]
\[x = 48\]
Finally, we verify the solution by substituting \(x = 48\) back into the original equation \(\sqrt{3x} - 6 = 6\):
\[\sqrt{3 \times 48} - 6 = \sqrt{144} - 6 = 12 - 6 = 6\]
Since the value 6 matches the right side of the original equation, the solution is correct. The condition \(x \geq 0\) is also satisfied as \(48 \geq 0\). Thus, the real solution for the equation is \(x = 48\). The final answer is \(\boxed{48}\).
Radical equations are common in physics and engineering, such as calculating the speed of an object based on its kinetic energy or analyzing signals in electrical engineering. For instance, if you are calculating the time it takes for a pendulum to swing or determining the distance an object has fallen based on its velocity, you often encounter square root expressions. Mastering the techniques to solve such equations, like isolating the root and squaring both sides, allows you to accurately model and solve these physical phenomena.