Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(x + 8) - √(x - 4) = 2

Verified step by step guidance

1. Start by isolating one of the square root terms. For example, rewrite the equation as \(\sqrt{{x + 8}} = \sqrt{{x - 4}} + 2\).
2. Square both sides of the equation to eliminate the square root on the left. This gives: \(\left(\sqrt{{x + 8}}\right)^2 = \left(\sqrt{{x - 4}} + 2\right)^2\).
3. Simplify both sides: the left side becomes \(x + 8\), and the right side expands using the formula \((a + b)^2 = a^2 + 2ab + b^2\) to \( (x - 4) + 4\sqrt{{x - 4}} + 4\).
4. Rearrange the equation to isolate the remaining square root term: \(x + 8 = x - 4 + 4\sqrt{{x - 4}} + 4\), then simplify to get \$4\sqrt{{x - 4}} = 8$.
5. Divide both sides by 4 to isolate the square root: \(\sqrt{{x - 4}} = 2\). Then square both sides again to solve for \(x\): \(x - 4 = 4\). Finally, solve for \(x\).

Key Concepts

Radical Equations

Radical equations involve variables inside a root, often a square root. To solve them, isolate the radical expression and then eliminate the root by raising both sides of the equation to the appropriate power, typically squaring for square roots.

Domain Restrictions

When dealing with radicals, especially square roots, the expression inside the root must be non-negative to produce real solutions. Identifying domain restrictions ensures that any proposed solutions are valid within the problem's context.

Checking Solutions

After solving radical equations, it is essential to substitute solutions back into the original equation to verify they do not produce extraneous results. Squaring both sides can introduce invalid solutions, so checking prevents incorrect answers.