Solve each equation. See Examples 4–6. √(4x+13) = 2x-1

Verified step by step guidance

1. Start by identifying the equation: \(\sqrt{4x + 13} = 2x - 1\). Notice that the square root expression must be equal to a non-negative value because the square root function outputs only non-negative results. So, first consider the domain where \$2x - 1 \geq 0$.
2. Square both sides of the equation to eliminate the square root. This gives: \((\sqrt{4x + 13})^2 = (2x - 1)^2\), which simplifies to \$4x + 13 = (2x - 1)^2$.
3. Expand the right side: \((2x - 1)^2 = (2x)^2 - 2 \cdot 2x \cdot 1 + 1^2 = 4x^2 - 4x + 1\). So the equation becomes \$4x + 13 = 4x^2 - 4x + 1$.
4. Bring all terms to one side to set the equation equal to zero: \$0 = 4x^2 - 4x + 1 - 4x - 13$. Simplify the right side to get a quadratic equation.
5. Solve the quadratic equation using the quadratic formula or factoring if possible. After finding the solutions, check each one in the original equation to ensure they do not produce extraneous solutions, especially considering the domain restriction from step 1.

Key Concepts

Solving Radical Equations

Radical equations involve variables inside a root, often a square root. To solve, isolate the radical expression and then eliminate the root by raising both sides to the appropriate power, typically squaring for square roots. This process can 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 possible values of the variable, so determining the domain before solving helps avoid invalid solutions and simplifies the solving process.

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 which are valid. Only solutions that satisfy the original equation are accepted.