Meaning Preservation Math Example 2

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

Example 2

medium
Squaring both sides of x=xβˆ’2\sqrt{x} = x - 2 can introduce extraneous solutions. Solve the equation and check which solutions are valid.

Solution

  1. 1
    Square both sides: x=(xβˆ’2)2=x2βˆ’4x+4x = (x-2)^2 = x^2 - 4x + 4.
  2. 2
    Rearrange: x2βˆ’5x+4=0x^2 - 5x + 4 = 0, factor: (xβˆ’1)(xβˆ’4)=0(x-1)(x-4) = 0, so x=1x = 1 or x=4x = 4.
  3. 3
    Check x=1x=1: LHS =1=1= \sqrt{1}=1; RHS =1βˆ’2=βˆ’1= 1-2=-1. 1β‰ βˆ’11 \ne -1 β€” extraneous, reject.
  4. 4
    Check x=4x=4: LHS =4=2= \sqrt{4}=2; RHS =4βˆ’2=2= 4-2=2. Valid.

Answer

x=4Β (only);Β x=1Β isΒ extraneousx = 4 \text{ (only); } x=1 \text{ is extraneous}
Squaring is not an equivalence-preserving operation β€” it can introduce solutions that do not satisfy the original equation. Always check solutions in the original equation after squaring.

About Meaning Preservation

Meaning preservation is the principle that valid mathematical transformations must maintain the truth and relationships of the original expression β€” changing form without changing content.

Learn more about Meaning Preservation β†’

More Meaning Preservation Examples

Example 1 easy

When solving [formula], list each algebraic step and explain why it preserves the meaning (solution

Example 3 easy

Which operation preserves the solution set of an equation: (a) multiply both sides by 3, (b) multipl

Example 4 medium

A student divides both sides of [formula] by [formula], getting [formula], so [formula]. Identify th

← All Meaning Preservation Examples Meaning Preservation Concept