Solve each equation. √(2x-5)=2+√(x-2)

Verified step by step guidance

1. Start by isolating one of the square root expressions to make the equation easier to handle. For example, write the equation as \(\sqrt{2x - 5} = 2 + \sqrt{x - 2}\).
2. To eliminate the square roots, square both sides of the equation. Remember that when you square a binomial like \((a + b)^2\), it expands to \(a^2 + 2ab + b^2\). So, square the right side carefully: \(\left(2 + \sqrt{x - 2}\right)^2\).
3. After squaring, simplify both sides of the equation. This will give you an expression without the square root on the left, but the right side will still contain a square root term.
4. Isolate the remaining square root term on one side of the equation again, then square both sides a second time to completely eliminate the square roots.
5. Once all square roots are removed, solve the resulting polynomial equation for \(x\). Finally, check your solutions by substituting them back into the original equation to ensure they do not produce extraneous results.

Key Concepts

Square Root Equations

Square root equations involve variables inside a radical sign. To solve them, isolate the square root on one side and then square both sides to eliminate the radical. This process may introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions

The domain of a square root function includes only values that make the radicand (expression inside the root) non-negative. For example, in √(2x-5), the expression 2x-5 must be ≥ 0. Identifying domain restrictions helps avoid invalid or extraneous solutions.

Solving Radical Equations by Squaring

When an equation has radicals on both sides, isolate one radical and square both sides to remove it. Repeat if necessary. After squaring, solve the resulting polynomial or linear equation, then verify solutions to exclude any that do not satisfy the original equation.