Solve $7+\sqrt{x}=18$ for $x$.
The equation to solve is given as \(7 + \sqrt{x} = 18\). The variable \(x\) must be a non-negative real number for the square root \(\sqrt{x}\) to be defined in the set of real numbers. Our goal is to isolate \(x\) and find its value.
To isolate the square root, we subtract 7 from both sides of the equation:
\[\sqrt{x} = 18 - 7\]
\[\sqrt{x} = 11\]
Now that the square root is isolated, we square both sides of the equation to eliminate the radical:
\[(\sqrt{x})^2 = 11^2\]
\[x = 121\]
Since \(x=121\) is positive, it satisfies the domain condition for the square root.
We substitute \(x = 121\) back into the original equation to verify our result:
\[7 + \sqrt{121} = 7 + 11 = 18\]
The result matches the original equation, so the solution is correct. The corresponding option is \(x = 121\).
Solving equations with radicals is fundamental in fields like physics and engineering. For example, if you are calculating the time it takes for an object to fall from a certain height, the distance is related to the square of time (\(d = \frac{1}{2}gt^2\)). If you know the distance and need to find the time, you must solve for \(t\) by taking a square root, just as we did here to find the value of \(x\).