Solve each equation. See Examples 4–6. √3x=√(5x+1)-1

Verified step by step guidance

1. Start with the given equation: \(\sqrt{3x} = \sqrt{5x + 1} - 1\).
2. Isolate one of the square root terms by adding 1 to both sides: \(\sqrt{3x} + 1 = \sqrt{5x + 1}\).
3. Square both sides of the equation to eliminate the square roots: \(\left(\sqrt{3x} + 1\right)^2 = \left(\sqrt{5x + 1}\right)^2\).
4. Expand the left side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \$3x + 2\sqrt{3x} + 1 = 5x + 1$.
5. Simplify the equation and isolate the remaining square root term, then square both sides again to solve for \(x\).

Key Concepts

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 simplify solving. Proper isolation helps prevent errors and makes the equation easier to handle.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation, called extraneous solutions. After solving, substitute each solution back into the original equation to verify its validity and discard any that do not work.