Solve the equation:
$\sqrt{x+20}=9$
To solve the equation \(\sqrt{x+20} = 9\), we first identify the variable \(x\) under the square root and the constraints for the equation to be valid. The expression inside the square root must be non-negative, meaning \(x + 20 \geq 0\), or \(x \geq -20\). Our goal is to isolate \(x\) by removing the square root operator.
To eliminate the square root, we square both sides of the equation:
\[(\sqrt{x+20})^2 = 9^2\]
This simplifies the equation to:
\[x + 20 = 81\]
Now, to isolate \(x\), we subtract 20 from both sides:
\[x = 81 - 20\]
\[x = 61\]
Finally, we verify the solution by substituting \(x = 61\) back into the original equation:
\[\sqrt{61+20} = \sqrt{81} = 9\]
Since the result is consistent with the original equation, \(61\) is the correct solution.
In engineering or physics, radical equations often appear when calculating distances, velocities, or signal strengths based on energy or area (which are related to the square of a variable). For instance, if the strength of a signal is related to the distance \(d\) from a transmitter by the formula \(\sqrt{d+20} = 9\), solving this equation helps engineers determine the precise distance from the source. Mastering these steps allows you to solve for unknown variables in various real-world models.