Question

Solve. $$ \sqrt{m-1}=\sqrt{m-\sqrt{m-4}} $$

   Solve.
$$
\sqrt{m-1}=\sqrt{m-\sqrt{m-4}}
$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 3rd Edition

Chapter 10, Problem 86

Instant Answer

Solved by Expert Narayan Hari

02/19/2021

Step 1

We can square both sides to get rid of the square roots. This gives us $m-1 = m - \sqrt{m-4}$. Show more…

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Solve. $$ \sqrt{m-1}=\sqrt{m-\sqrt{m-4}} $$

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Key Concepts

Properties of Square Roots

Square roots represent the non-negative value that, when multiplied by itself, returns the original number. Understanding that the square root function is defined only for non-negative inputs is crucial, especially when the square root appears on both sides of an equation, ensuring that the manipulations follow the function's inherent properties.

Domain Restrictions

When solving equations that involve square roots, it is important to consider the domains of the functions involved. All expressions under a square root must be non-negative, which places constraints on the variable. Evaluating these restrictions is a critical step in ensuring that any proposed solution is valid within the context of the original problem.

Squaring Both Sides of an Equation

Squaring both sides of an equation is a common technique used to eliminate square roots to simplify the equation. However, this operation is not always reversible because it can introduce solutions that do not satisfy the original equation. Thus, while it transforms the equation into a more manageable form, care must be taken to validate the solutions afterward.

Extraneous Solutions

Extraneous solutions are those that arise from the process of solving an equation but do not satisfy the original equation. They are particularly common when both sides of an equation are squared, as such operations can introduce invalid solutions. It is essential to substitute any potential solutions back into the original equation to verify their legitimacy.

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