In Exercises 1–38, solve each radical equation.(x - 2)¹/² + 8 = 6

Verified step by step guidance

1. Start by isolating the radical expression on one side of the equation. Subtract 8 from both sides: \((x - 2)^{1/2} = 6 - 8\).
2. Simplify the right side of the equation: \((x - 2)^{1/2} = -2\).
3. To eliminate the square root, square both sides of the equation: \(((x - 2)^{1/2})^2 = (-2)^2\).
4. Simplify both sides: \(x - 2 = 4\).
5. Solve for \(x\) by adding 2 to both sides: \(x = 4 + 2\).

Key Concepts

Radical Equations

Radical equations are equations that involve a variable within a radical (square root, cube root, etc.). To solve these equations, one typically isolates the radical on one side and then raises both sides of the equation to the power that eliminates the radical. This process may introduce extraneous solutions, so it's important to check all potential solutions in the original equation.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable by itself on one side. In the context of radical equations, this often means moving other terms to the opposite side before applying operations to eliminate the radical, which simplifies the equation and makes it easier to solve.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the process of solving an equation but do not satisfy the original equation. This is particularly common in radical equations, where squaring both sides can introduce solutions that are not valid. Therefore, it is crucial to substitute any found solutions back into the original equation to verify their validity.