Solve each radical equation in Exercises 11–30. Check all proposed solutions. $\sqrt{1+4\sqrt{x}}=1+\sqrt{x}$

Verified step by step guidance

1. Start by letting \( y = \sqrt{x} \). This substitution simplifies the equation and helps avoid dealing with nested radicals directly.
2. Rewrite the original equation \( \sqrt{1 + 4\sqrt{x}} = 1 + \sqrt{x} \) as \( \sqrt{1 + 4y} = 1 + y \) using the substitution.
3. Square both sides of the equation to eliminate the square root on the left side: \( (\sqrt{1 + 4y})^2 = (1 + y)^2 \), which simplifies to \( 1 + 4y = (1 + y)^2 \).
4. Expand the right side: \( (1 + y)^2 = 1 + 2y + y^2 \). So the equation becomes \( 1 + 4y = 1 + 2y + y^2 \).
5. Bring all terms to one side to form a quadratic equation: \( 1 + 4y - 1 - 2y - y^2 = 0 \), which simplifies to \( y^2 - 2y = 0 \). Then solve for \( y \) by factoring or using the quadratic formula.

Key Concepts

Radical Equations

Radical equations involve variables inside a root, such as square roots. Solving them often requires isolating the radical expression and then eliminating the root by raising both sides to the appropriate power. This process can introduce extraneous solutions, so checking all proposed solutions is essential.

Nested Radicals

Nested radicals occur when a radical expression contains another radical inside it, like √(1 + 4√x). Understanding how to simplify or manipulate nested radicals is crucial for solving such equations, often by substitution or careful algebraic manipulation.

Extraneous Solutions and Verification

When solving radical equations, raising both sides to a power can introduce solutions that do not satisfy the original equation. Therefore, every solution must be substituted back into the original equation to verify its validity and discard any extraneous solutions.