Solve the square root equation $\sqrt{2 x-3}=5$.
We are asked to solve for \(x\) in the equation \(\sqrt{2x - 3} = 5\). The goal is to isolate \(x\) by removing the square root, solving the resulting linear equation, and verifying that the solution satisfies the original expression. We are given the potential options \(x = 11, x = 1, x = 4, x = 14\).
To eliminate the square root, we square both sides of the equation:
\[(\sqrt{2x - 3})^2 = 5^2\]
This simplifies the left side by removing the radical, and squares the right side:
\[2x - 3 = 25\]
Now, we solve for \(x\) by first isolating the term with \(x\):
\[2x = 25 + 3\]
\[2x = 28\]
Dividing both sides by 2 gives:
\[x = \frac{28}{2} = 14\]
Finally, we verify our solution by substituting \(x = 14\) back into the original equation:
\[\sqrt{2(14) - 3} = \sqrt{28 - 3} = \sqrt{25} = 5\]
Since the result matches the right-hand side of the original equation, \(x = 14\) is indeed the correct solution. Among the given options, this corresponds to the last choice. The correct value is \(\boxed{14}\).
Square root equations are often used in physics to relate the velocity of an object to the energy applied to it or the distance it has traveled. For example, if you know the potential energy applied to a spring, you can calculate the speed at which it might propel an object. Solving these equations helps engineers and scientists predict outcomes in mechanics, ensuring safety and precision in systems like car suspensions or sports equipment design.