What is the solution of $\sqrt{x+2}-15=-3$ ?
The equation given is \(\sqrt{x+2} - 15 = -3\). Our first goal is to isolate the square root expression to solve for \(x\). The square root function \(\sqrt{u}\) is defined for \(u \ge 0\), which implies \(x+2 \ge 0\), or \(x \ge -2\). Furthermore, the principal square root \(\sqrt{u}\) always yields a non-negative value, \(\sqrt{u} \ge 0\). We will keep these conditions in mind throughout our solution process.
To solve the equation \(\sqrt{x+2} - 15 = -3\), we add 15 to both sides to isolate the radical:
\[\sqrt{x+2} = -3 + 15\]
\[\sqrt{x+2} = 12\]
Since 12 is a non-negative number, this equation is well-defined and consistent with the properties of the square root function.
Now that we have \(\sqrt{x+2} = 12\), we square both sides of the equation to eliminate the radical:
\[(\sqrt{x+2})^2 = 12^2\]
\[x + 2 = 144\]
Subtracting 2 from both sides gives the value of \(x\):
\[x = 144 - 2\]
\[x = 142\]
We must verify that \(x = 142\) satisfies the original equation \(\sqrt{x+2} - 15 = -3\):
\[\sqrt{142+2} - 15 = \sqrt{144} - 15\]
\[= 12 - 15 = -3\]
The solution \(x = 142\) is consistent with the original equation and satisfies the constraint \(x \ge -2\). Comparing this to the given options, we find that \(x = 142\) is indeed the correct solution.
In engineering and physics, radical equations often arise when relating speed, distance, and acceleration, or when analyzing signal power attenuation. For instance, calculating the required length of a structural component to ensure a specific level of tension involves solving radical equations where the physical constraints (like length must be positive) determine the validity of the result. Understanding how to isolate variables and verify solutions ensures that you select the correct physical model for real-world design.