Solve each equation. (√x+2)-x = 2

Verified step by step guidance

1. Start by rewriting the equation to isolate the square root term: \(\sqrt{x + 2} - x = 2\).
2. Add \(x\) to both sides to get \(\sqrt{x + 2} = x + 2\).
3. Square both sides of the equation to eliminate the square root: \((\sqrt{x + 2})^2 = (x + 2)^2\), which simplifies to \(x + 2 = (x + 2)^2\).
4. Expand the right side: \(x + 2 = x^2 + 4x + 4\).
5. Bring all terms to one side to form a quadratic equation: \$0 = x^2 + 4x + 4 - x - 2\(, which simplifies to \)0 = x^2 + 3x + 2$. Then solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Key Concepts

Square Root and Radicals

A square root of a number x is a value that, when multiplied by itself, gives x. In equations, radicals like √x represent these roots and require careful handling, especially when isolating the radical term before solving.

Isolating the Radical

To solve equations involving square roots, first isolate the radical expression on one side. This step is crucial because it allows you to eliminate the square root by squaring both sides, simplifying the equation into a polynomial form.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that don't satisfy the original equation. After solving, substitute the solutions back into the original equation to verify which are valid and discard any extraneous ones.