Equivalence Transformation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Why is squaring both sides of an equation NOT always an equivalence transformation?

Solution

  1. 1
    Start with x=3x = 3. Square both sides: x2=9x^2 = 9.
  2. 2
    But x2=9x^2 = 9 has solutions x=3x = 3 and x=−3x = -3.
  3. 3
    Squaring introduced an extra solution (x=−3x = -3) that doesn't satisfy the original.
  4. 4
    Therefore squaring can add extraneous solutions.

Answer

Squaring can introduce extraneous solutions.
Not all operations are equivalence transformations. Squaring both sides can enlarge the solution set, which is why we must always check solutions when solving radical or absolute value equations.

About Equivalence Transformation

Operations applied to both sides of an equation that transform its form while leaving its solution set completely unchanged.

Learn more about Equivalence Transformation →

More Equivalence Transformation Examples

Example 1 easy

Starting from [formula], apply an equivalence transformation to solve.

Example 3 easy

What equivalence transformation converts [formula] to [formula]?

Example 4 medium

Why must we NOT multiply both sides of an equation by 0?

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