Solve: √(6x - 2) = √(2x + 3) - √(4x - 1).

Verified step by step guidance

1. Step 1: Start by isolating one of the square root terms. Add √(4x - 1) to both sides of the equation to get: √(6x - 2) + √(4x - 1) = √(2x + 3).
2. Step 2: To eliminate the square roots, square both sides of the equation. On the left-hand side, use the binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2. This gives: (√(6x - 2))^2 + 2√(6x - 2)√(4x - 1) + (√(4x - 1))^2 = (√(2x + 3))^2.
3. Step 3: Simplify each term. The squares of the square roots simplify to their respective radicands: 6x - 2 + 2√((6x - 2)(4x - 1)) + 4x - 1 = 2x + 3.
4. Step 4: Combine like terms on the left-hand side. This results in: (6x - 2) + (4x - 1) + 2√((6x - 2)(4x - 1)) = 2x + 3. Simplify further to get: 10x - 3 + 2√((6x - 2)(4x - 1)) = 2x + 3.
5. Step 5: Isolate the remaining square root term by subtracting (10x - 3) from both sides: 2√((6x - 2)(4x - 1)) = 2x + 3 - (10x - 3). Simplify the right-hand side and proceed to square both sides again to eliminate the square root. Continue solving for x by simplifying and isolating x.

Key Concepts

Square Roots

Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. In the equation, the square root functions indicate that we are dealing with non-negative values, as square roots of real numbers are defined only for non-negative inputs. Understanding how to manipulate square roots is essential for solving equations involving them.

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, allowing for easier manipulation and solution. In the given problem, isolating the square root terms will help simplify the equation and make it solvable.

Squaring Both Sides

Squaring both sides of an equation is a common method used to eliminate square roots. When both sides of an equation are squared, it can simplify the equation, but it is crucial to check for extraneous solutions afterward, as squaring can introduce solutions that do not satisfy the original equation. This step is particularly relevant in the context of the given problem.