The method
- Isolate the root on one side.
- Square both sides. The root goes, and what is left is usually a quadratic equation.
- Solve it, for instance with the quadratic formula or by factorising.
- Check every root in the original equation. A root that does not hold is a false (extraneous) root and is rejected.
An example with a false root: √(x + 7) = x + 1
Squaring gives x + 7 = x² + 2x + 1, so x² + x − 6 = 0, with the roots x = 2 and x = −3.
- x = 2: the left side is √9 = 3 and the right side 2 + 1 = 3. They agree, so x = 2 is a solution.
- x = −3: the left side is √4 = 2 but the right side −3 + 1 = −2. They differ, so x = −3 is a false root.
The answer is x = 2.
An example without one: √(3x + 4) = x + 2
Squaring gives 3x + 4 = x² + 4x + 4, so x² + x = 0, with the roots x = 0 and x = −1. Checking: √4 = 2 = 0 + 2 and √1 = 1 = −1 + 2. Both hold, so the equation has two solutions.
Why false roots appear
Squaring turns a = b into a² = b², and the second also holds when a = −b. A square root is never negative, so a root that makes the right side negative cannot solve the original equation. That is the root the check catches. The step a = b ⇒ a² = b² is an implication, not an equivalence.
Common mistakes
- Skipping the check. It is what tells a solution from a false root.
- Checking in the squared equation. Both roots hold there; the check is made in the equation as it was given.
- Squaring term by term. (x + 1)² is x² + 2x + 1, not x² + 1.
- Squaring before the root is isolated. √(x + 7) + 2 = x + 3 is squared only after 2 has been subtracted from both sides.
See also
The quadratic formula for the quadratic equation, and fractional exponents for a root written as a power.