Solve each equation. See Examples 4–6. 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. Square both sides of the equation to eliminate the square root: \(x^2 = (\sqrt{2x + 3})^2\) 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, then check each solution in the original equation to avoid extraneous roots.

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. Squaring transforms the equation into a polynomial form that is easier to solve.

Checking for Extraneous Solutions

After solving an equation involving radicals, some solutions may not satisfy the original equation due to the squaring step. Substituting each solution back into the original equation verifies its validity. This ensures only true solutions are accepted.