Solve each equation. √x-√(x-12)=2

Verified step by step guidance

1. Identify the equation given: \(\sqrt{x} - \sqrt{x - 12} = 2\). Our goal is to solve for \(x\).
2. Isolate one of the square root terms to one side. For example, add \(\sqrt{x - 12}\) to both sides to get \(\sqrt{x} = 2 + \sqrt{x - 12}\).
3. Square both sides of the equation to eliminate the square root on the left. This gives: \(\left(\sqrt{x}\right)^2 = \left(2 + \sqrt{x - 12}\right)^2\).
4. Simplify both sides: the left side becomes \(x\), and the right side expands using the formula \((a + b)^2 = a^2 + 2ab + b^2\) to \(4 + 4\sqrt{x - 12} + (x - 12)\).
5. Rearrange the equation to isolate the remaining square root term, then square both sides again to eliminate it. Finally, solve the resulting polynomial equation for \(x\) and check all solutions in the original equation to avoid extraneous roots.

Key Concepts

Square Roots and Radicals

Square roots are the inverse operation of squaring a number, represented by the radical symbol √. Understanding how to manipulate and simplify expressions involving square roots is essential, especially when isolating terms or combining radicals.

Isolating and Solving Radical Equations

Solving equations with radicals often requires isolating the radical expression on one side and then squaring both sides to eliminate the square root. This process may need to be repeated and requires careful handling to avoid extraneous solutions.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. Therefore, it is crucial to substitute all potential solutions back into the original equation to verify their validity.