Solve each equation. See Examples 4–6. √(x+5) + 2 = √(x-1)

Verified step by step guidance

1. Isolate one of the square root expressions on one side of the equation. For example, subtract 2 from both sides to get: \(\sqrt{{x+5}} = \sqrt{{x-1}} - 2\).
2. Square both sides of the equation to eliminate the square roots. Remember to apply the formula \((a - b)^2 = a^2 - 2ab + b^2\) when squaring the right side: \(\left(\sqrt{{x+5}}\right)^2 = \left(\sqrt{{x-1}} - 2\right)^2\).
3. Simplify both sides after squaring. The left side becomes \(x + 5\). The right side expands to \((x - 1) - 4\sqrt{{x-1}} + 4\).
4. Rearrange the equation to isolate the remaining square root term. Move all terms without the square root to one side and keep the square root term on the other side.
5. Square both sides again to eliminate the remaining square root. Then solve the resulting quadratic equation for \(x\). Finally, check your solutions by substituting them back into the original equation to avoid extraneous solutions.

Key Concepts

Square Root Functions and Their Properties

Square root functions involve expressions under a radical sign, representing the principal (non-negative) root. Understanding how to manipulate and isolate square roots is essential for solving equations containing radicals.

Isolating Radicals and Squaring Both Sides

To solve equations with square roots, isolate the radical expression on one side and then square both sides to eliminate the square root. This step must be done carefully to avoid introducing extraneous solutions.

Checking for Extraneous Solutions

After solving the squared equation, substitute solutions back into the original equation to verify their validity. Squaring can introduce extraneous solutions that do not satisfy the original radical equation.