Solve each equation. √(4x+13) = 2x-1

Verified step by step guidance

1. Start by understanding the equation: \(\sqrt{4x + 13} = 2x - 1\). The goal is to solve for \(x\).
2. Since the equation involves a square root, isolate the square root expression on one side (which it already is) and then square both sides to eliminate the square root. This gives: \(\left(\sqrt{4x + 13}\right)^2 = (2x - 1)^2\).
3. Simplify both sides: the left side becomes \$4x + 13\(, and the right side expands using the formula \)(a - b)^2 = a^2 - 2ab + b^2\(, so \)(2x - 1)^2 = 4x^2 - 4x + 1$.
4. Set up the resulting quadratic equation by equating both sides: \$4x + 13 = 4x^2 - 4x + 1\(. Then, move all terms to one side to get \)0 = 4x^2 - 4x + 1 - 4x - 13$.
5. Simplify the quadratic equation and solve for \(x\) using factoring, completing the square, or the quadratic formula. Remember to check your solutions in the original equation because squaring both sides can introduce extraneous solutions.

Key Concepts

Solving Radical Equations

Radical equations involve variables inside a root, such as a square root. To solve them, isolate the radical expression and then eliminate the root by raising both sides of the equation to the appropriate power, typically squaring for square roots. This process may introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions

When dealing with square roots, the expression inside the root must be non-negative to produce real numbers. This restriction limits the domain of possible solutions. Before solving, identify values of the variable that keep the radicand (expression under the root) greater than or equal to zero to ensure valid solutions.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. After finding potential solutions, substitute them back into the original equation to verify their validity. Only solutions that satisfy the original equation are accepted.