Solve each equation. See Examples 4–6. √x+2=√(4+7√x)

Verified step by step guidance

1. Start by rewriting the equation to clearly identify the expressions under the square roots: \(\sqrt{x} + 2 = \sqrt{4 + 7\sqrt{x}}\).
2. Isolate one of the square root terms to prepare for squaring both sides. For example, keep the equation as is and then square both sides to eliminate the square roots: \(\left(\sqrt{x} + 2\right)^2 = \left(\sqrt{4 + 7\sqrt{x}}\right)^2\).
3. Expand the left side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \(\left(\sqrt{x}\right)^2 + 2 \cdot \sqrt{x} \cdot 2 + 2^2 = 4 + 7\sqrt{x}\).
4. Simplify the expanded terms: \(x + 4\sqrt{x} + 4 = 4 + 7\sqrt{x}\).
5. Next, subtract 4 from both sides to simplify and then isolate terms involving \(\sqrt{x}\): \(x + 4\sqrt{x} = 7\sqrt{x}\). Then, bring all terms to one side to set the equation to zero and solve for \(\sqrt{x}\).

Key Concepts

Square Roots and Radicals

Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify expressions involving square roots is essential, especially when they appear on both sides of an equation.

Isolating Variables in Radical Equations

To solve equations with radicals, it is important to isolate the radical expression before squaring both sides. This helps eliminate the square root and simplifies the equation, but care must be taken to check for extraneous solutions.

Checking for Extraneous Solutions

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