Solve each equation. √2x=√(3x+12)-2

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Start by rewriting the equation to clearly identify the square root expressions: \(\sqrt{2x} = \sqrt{3x + 12} - 2\).

Isolate one of the square root terms on one side. In this case, add 2 to both sides to get: \(\sqrt{2x} + 2 = \sqrt{3x + 12}\).

Square both sides of the equation to eliminate the square roots. This means squaring \(\left(\sqrt{2x} + 2\right)\) and setting it equal to \(\left(\sqrt{3x + 12}\right)^2\).

After squaring, expand the left side using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = \sqrt{2x}\) and \(b = 2\). The right side simplifies to \(3x + 12\).

Simplify the resulting equation and solve the quadratic equation for \(x\). Remember to check your solutions by substituting them back into the original equation to avoid extraneous roots.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Isolating Variables

Isolating the variable means manipulating the equation to get the variable alone on one side. This step is crucial before squaring both sides to avoid complicating the equation and to ensure accurate solutions. It often involves adding, subtracting, or factoring terms.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that don't satisfy the original equation. After finding potential solutions, substitute them back into the original equation to verify their validity. Only solutions that satisfy the original equation are accepted.

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