Solve each equation. √x-√(x-5)=1

Verified step by step guidance

1. Identify the equation to solve: \(\sqrt{x} - \sqrt{x - 5} = 1\).
2. Isolate one of the square root terms. For example, add \(\sqrt{x - 5}\) to both sides to get \(\sqrt{x} = 1 + \sqrt{x - 5}\).
3. Square both sides of the equation to eliminate the square root on the left: \(\left(\sqrt{x}\right)^2 = \left(1 + \sqrt{x - 5}\right)^2\).
4. Expand the right side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \(x = 1 + 2\sqrt{x - 5} + (x - 5)\).
5. Simplify the equation and isolate the remaining square root term, then square both sides again to solve for \(x\). Finally, check your solutions in the original equation to avoid extraneous roots.

Key Concepts

Square Roots and Radicals

Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify expressions involving square roots is essential, especially when they appear in equations. Recognizing domain restrictions, such as the radicand being non-negative, is also crucial.

Isolating Radical Expressions

To solve equations with square roots, isolate one radical expression on one side of the equation. This step allows you to eliminate the square root by squaring both sides, simplifying the equation into a polynomial form that is easier to solve.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. After solving, substitute the solutions back into the original equation to verify their validity and discard any extraneous solutions.