Squaring both sides of an equation is a common practice in algebra, but it’s important to understand the conditions under which this operation is valid. The main reason we can square both sides is because the operation is equal and maintains the relationship between the two sides of the equation, as long as we take care of certain conditions.
When we have an equation of the form A = B, squaring both sides gives us A² = B². This transformation is based on the assumption that the original equation holds true, meaning that whatever values A and B represent, they must be equal.
However, squaring both sides can lead to extraneous solutions. For example, if we squared both sides of the equation -2 = 2, we would get (-2)² = 2², which simplifies to 4 = 4, a true statement. However, the original equation is false since -2 does not equal 2.
Thus, while you can square both sides of an equation, always ensure to check for extraneous solutions by substituting back into the original equation to validate your results. In summary, squaring is a valid operation under the right circumstances, but it’s crucial to be aware of its limitations.