Solve each equation. √x+2-x = 2

Verified step by step guidance

1. Start by isolating the square root term on one side of the equation. The given equation is \(\sqrt{x} + 2 - x = 2\). Subtract 2 from both sides to get \(\sqrt{x} - x = 0\).
2. Rewrite the equation as \(\sqrt{x} = x\) to express the square root term equal to \(x\).
3. Square both sides of the equation to eliminate the square root. This gives \((\sqrt{x})^2 = x^2\), which simplifies to \(x = x^2\).
4. Rearrange the equation to standard quadratic form: \(x^2 - x = 0\).
5. Factor the quadratic equation: \(x(x - 1) = 0\). Set each factor equal to zero to find possible solutions: \(x = 0\) or \(x = 1\). Remember to check these solutions in the original equation to avoid extraneous roots.

Key Concepts

Square Root and Radicals

A square root of a number is a value that, when multiplied by itself, gives the original number. In equations, radicals like √x represent the principal (non-negative) root. Understanding how to manipulate and isolate radicals is essential for solving equations involving square roots.

Isolating the Radical Expression

To solve equations with radicals, first isolate the square root term on one side of the equation. This step allows you to eliminate the radical by squaring both sides, simplifying the equation into a polynomial form that is easier to solve.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation. After finding potential solutions, substitute them back into the original equation to verify which are valid.