Verified Answer :

  • Isolate the radical term by subtracting 3 from both sides: \(\sqrt{x} = 9\).
  • Square both sides to eliminate the square root: \(x = 9^2 = 81\).
  • Verify the solution by substituting back into the original equation: \(\sqrt{81} + 3 = 9 + 3 = 12\).
  • The solution is \(\boxed{81}\).

Explanation

  1. Isolating the Radical Term

To solve the equation \(\sqrt{x} + 3 = 12\), we start by isolating the term with the square root. We subtract 3 from both sides of the equation:
\[\sqrt{x} = 12 - 3\]
\[\sqrt{x} = 9\]

  1. Solving for x

To eliminate the square root, we square both sides of the equation:
\[(\sqrt{x})^2 = 9^2\]
\[x = 81\]

  1. Verification and Final Result

Finally, we check the solution by substituting \(x = 81\) back into the original equation:
\[\sqrt{81} + 3 = 9 + 3 = 12\]
Since the equation holds true, \(x = 81\) is the correct solution. Comparing this with the given options (225, 81, 3, and no solution), we see that 81 is indeed the correct choice.

Examples

Solving for an unknown variable within a radical equation is a fundamental skill in algebra, useful in fields like engineering and physics. For example, if you are calculating the time \(t\) it takes for an object to fall a certain distance \(d\) under gravity, you might encounter an equation like \(\sqrt{d} = k \times t\). Understanding how to isolate and solve for variables helps you analyze physical phenomena, determine safety thresholds, or calculate resource requirements in real-world projects.

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