Solve each equation. √4x-2 = √3x+1

Verified step by step guidance

1. Start by writing the given equation clearly: \(\sqrt{4x - 2} = \sqrt{3x + 1}\).
2. To eliminate the square roots, square both sides of the equation: \((\sqrt{4x - 2})^2 = (\sqrt{3x + 1})^2\).
3. Simplify both sides after squaring: \$4x - 2 = 3x + 1$.
4. Isolate the variable \(x\) by subtracting \$3x\( from both sides and adding \(2\) to both sides: \)4x - 3x = 1 + 2$.
5. Solve the resulting linear equation for \(x\): \(x = 3\).

Key Concepts

Square Roots and Radicals

Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify square root expressions is essential for solving equations involving radicals.

Isolating and Squaring Both Sides

To solve equations with square roots, isolate the radical expressions and then square both sides to eliminate the square roots. This step often transforms the equation into a polynomial form that is easier to solve.

Checking for Extraneous Solutions

Squaring both sides can introduce solutions that do not satisfy the original equation. It is important to substitute solutions back into the original equation to verify their validity and discard any extraneous roots.