Solve each equation. See Examples 4–6. √(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 to one side of the equation. 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 and Radicals

A square root of a number is a value that, when multiplied by itself, gives the original number. Radicals involve roots and are often expressed with the √ symbol. Understanding how to manipulate and simplify expressions with square roots is essential for solving equations involving radicals.

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. Care must be taken to avoid extraneous solutions introduced by squaring.

Checking for Extraneous Solutions

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