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 the Radical Expression

To solve equations with square roots, isolate the radical on one side before squaring both sides. This step helps eliminate the square root and simplifies the equation into a solvable algebraic form.

Checking for Extraneous Solutions

Squaring both sides can introduce solutions that don't satisfy the original equation. Always substitute solutions back into the original equation to verify their validity and discard any extraneous roots.