Solve each equation. See Examples 4–6. √2x=√(3x+12)-2

Verified step by step guidance

1. Start with the given equation: \(\sqrt{2x} = \sqrt{3x + 12} - 2\).
2. Isolate one of the square root terms to prepare for squaring. Add 2 to both sides to get: \(\sqrt{2x} + 2 = \sqrt{3x + 12}\).
3. Square both sides of the equation to eliminate the square roots: \(\left(\sqrt{2x} + 2\right)^2 = \left(\sqrt{3x + 12}\right)^2\).
4. Expand the left side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \(\left(\sqrt{2x}\right)^2 + 2 \times \sqrt{2x} \times 2 + 2^2 = 3x + 12\).
5. Simplify the terms: \$2x + 4\sqrt{2x} + 4 = 3x + 12\(. Then, isolate the radical term \)4\sqrt{2x}$ on one side to prepare for another squaring step.

Key Concepts

Solving Radical Equations

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

Domain Restrictions for Radicals

The expression inside a square root must be non-negative because the square root of a negative number is not real. When solving equations with radicals, determine the domain by setting the radicand ≥ 0 to ensure all solutions are valid within the real numbers.

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 ones are true solutions and discard any extraneous ones.