Solve each equation. √4x-2 = √3x+1

Verified step by step guidance

1. Step 1: Start by squaring both sides of the equation to eliminate the square roots. This gives you \((\sqrt{4x-2})^2 = (\sqrt{3x+1})^2\).
2. Step 2: Simplify both sides of the equation. The left side becomes \(4x - 2\) and the right side becomes \(3x + 1\).
3. Step 3: Set the simplified expressions equal to each other: \(4x - 2 = 3x + 1\).
4. Step 4: Solve for \(x\) by isolating it on one side of the equation. Subtract \(3x\) from both sides to get \(x - 2 = 1\).
5. Step 5: Add 2 to both sides to solve for \(x\), resulting in \(x = 3\).

Key Concepts

Square Roots

Square roots are mathematical operations that determine a number which, when multiplied by itself, gives the original number. In the equation √4x-2 = √3x+1, understanding how to manipulate square roots is essential for isolating variables and solving the equation. It is important to remember that squaring both sides of an equation can eliminate the square roots but may introduce extraneous solutions.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable of interest on one side and all other terms on the opposite side. In the context of the given equation, isolating x will help in simplifying the problem and finding the solution effectively.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the process of solving an equation but do not satisfy the original equation. When squaring both sides of an equation, it is crucial to check each potential solution in the original equation to ensure it is valid. This concept is particularly relevant in the given problem, as squaring the square roots can lead to additional, non-valid solutions.