Solve each equation. See Examples 4–6. √4x-x+3=0

Verified step by step guidance

1. Start by isolating the square root expression on one side of the equation: \(\sqrt{4x} - x + 3 = 0\). Rearrange it to get \(\sqrt{4x} = x - 3\).
2. Since the square root function outputs only non-negative values, consider the domain restriction: \$4x \geq 0\(, which implies \)x \geq 0\(. Also, because \)\sqrt{4x} = x - 3\(, the right side must be non-negative, so \)x - 3 \geq 0\(, or \)x \geq 3$.
3. Next, square both sides of the equation to eliminate the square root: \((\sqrt{4x})^2 = (x - 3)^2\), which simplifies to \$4x = (x - 3)^2$.
4. Expand the right side: \$4x = x^2 - 6x + 9\(. Then, bring all terms to one side to form a quadratic equation: \)0 = x^2 - 6x + 9 - 4x\(, which simplifies to \)x^2 - 10x + 9 = 0$.
5. Solve the quadratic equation \(x^2 - 10x + 9 = 0\) using the quadratic formula or factoring. After finding the solutions, check each one against the domain restrictions and the original equation to discard any extraneous solutions.

Key Concepts

Solving Radical Equations

Radical equations involve variables inside a root, such as a square root. To solve them, isolate the radical expression and then square both sides to eliminate the root. This process may introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions for Square Roots

The expression inside a square root must be non-negative because the square root of a negative number is not real. When solving equations with square roots, determine the domain by setting the radicand greater than or equal to zero to ensure valid solutions.

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 ones are valid and discard any extraneous solutions.