Solve each equation. x - √(2x+3) = 0

Verified step by step guidance

1. Start with the given equation: \(x - \sqrt{2x + 3} = 0\).
2. Isolate the square root term by adding \(\sqrt{2x + 3}\) to both sides: \(x = \sqrt{2x + 3}\).
3. To eliminate the square root, square both sides of the equation: \(x^2 = (\sqrt{2x + 3})^2\) which simplifies to \(x^2 = 2x + 3\).
4. Rewrite the equation in standard quadratic form by subtracting \$2x + 3\( from both sides: \)x^2 - 2x - 3 = 0$.
5. Solve the quadratic equation \(x^2 - 2x - 3 = 0\) using factoring, completing the square, or the quadratic formula to find the possible values of \(x\).

Key Concepts

Isolating the Variable

Isolating the variable means rearranging the equation to have the variable on one side alone. This step simplifies solving by making it easier to apply operations like squaring or factoring. For example, rewriting x - √(2x+3) = 0 as x = √(2x+3) helps in the next steps.

Solving Equations Involving Square Roots

Equations with square roots often require squaring both sides to eliminate the radical. This process can introduce extraneous solutions, so it's important to check all solutions in the original equation. For instance, squaring x = √(2x+3) leads to x² = 2x + 3.

Checking for Extraneous Solutions

After solving, substitute solutions back into the original equation to verify their validity. Squaring both sides can create solutions that don't satisfy the original equation, called extraneous solutions. This step ensures only true solutions are accepted.