Solve each equation. √2x-x+4=0

Verified step by step guidance

1. Start by rewriting the equation to clearly identify the terms: \(\sqrt{2x} - x + 4 = 0\).
2. Isolate the square root term on one side of the equation: \(\sqrt{2x} = x - 4\).
3. Square both sides of the equation to eliminate the square root: \((\sqrt{2x})^2 = (x - 4)^2\), which simplifies to \$2x = (x - 4)^2$.
4. Expand the right side using the binomial formula: \$2x = x^2 - 8x + 16$.
5. Rearrange the equation to standard quadratic form by bringing all terms to one side: \$0 = x^2 - 8x + 16 - 2x\(, which simplifies to \)0 = x^2 - 10x + 16$.

Key Concepts

Square Root Equations

Square root equations involve variables inside a radical sign. To solve them, isolate the square root expression 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.

Isolating Variables

Isolating the variable means rearranging the equation so that the variable appears alone on one side. This step is crucial before squaring both sides to avoid complicating the equation and to simplify solving for the variable.

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 which are valid and discard any extraneous ones.