Solve each equation.√(2x+5)-√(x+2)=1

Verified step by step guidance

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

Key Concepts

Square Root Functions and Radicals

Square root functions involve expressions under a radical sign (√). Understanding how to manipulate and simplify radicals is essential, especially when isolating terms or combining like radicals in an equation.

Isolating Radicals and Squaring Both Sides

To solve equations with square roots, isolate one radical on one side and then square both sides to eliminate the square root. This process may need to be repeated if multiple radicals are present, but be cautious of introducing extraneous solutions.

Checking for Extraneous Solutions

Squaring both sides can introduce solutions that do not satisfy the original equation. Always substitute your solutions back into the original equation to verify which are valid and discard any extraneous ones.