How do extraneous solutions arise from radical equations?
1 Answer
In general, extraneous solutions arise when we perform non-invertible operations on both sides of an equation. (That is, they sometimes arise, but not always.)
Squaring (or raising to any other even power) is a non-invertible operation. Solving equations involving square roots involves squaring both sides of an equation.
Example 1 : To show the idea:
The equations:
Square both sides of
Square both sides of
which is equivalent to
and, rewriting the left,
So the solution set is
This time, it is
Example2 : Extraneous solution.
Solve
Subtracting
Squaring (!) gives
This requires,
Factoring to get
finds the solution set to be
Checking these reveals that
Example 3 : No extraneous solution.
Solve
Squaring (!) gives,
Which leads to