Squaring to Solve
To solve an equation where the variable is inside a radical, you must isolate the radical and then square both sides.
Warning: Squaring can create "Extraneous Solutions"—answers that work in the squared equation but not in the original. Always check your answers.
Worked Examples
Example 1: Basic Root
Solve: \(\sqrt{x + 5} = 4\)
- Square both sides: \(x + 5 = 16\).
- Subtract 5: \(x = 11\).
- Check: \(\sqrt{11+5} = \sqrt{16} = 4\). Correct.
- Result: \(x = 11\)
Example 2: Two Steps
Solve: \(2\sqrt{x} - 6 = 0\)
- Isolate radical: \(2\sqrt{x} = 6 \to \sqrt{x} = 3\).
- Square sides: \(x = 9\).
- Result: \(x = 9\)
Example 3: Extraneous Solution
Solve: \(\sqrt{x} = -2\)
- Square sides: \(x = 4\).
- Check: \(\sqrt{4} = 2\), NOT -2.
- Result: No Solution (The square root of a real number is always positive).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we define the "Magnitude" of a wave as \(|\psi| = \sqrt{\psi^* \psi}\). If we want to solve for the physical probability current, we often have to "square" these expressions to get rid of the radicals. This lesson teaches you that squaring is a powerful way to unlock information trapped inside a root, but it also warns you that math can sometimes give you "ghost" answers that don't represent physical reality. Checking your work against physical constraints is the mark of a true physicist.