## Lesson 8: Solving Equations Involving Radicals Solving radical equations means finding the value of a variable that appears inside or alongside a square root. These often show up in rearranged electrical formulas and require careful steps to isolate the variable. ### General Approach To solve equations with square roots: 1. Isolate the radical expression 2. Square both sides to eliminate the square root 3. Solve the resulting equation 4. Check your answer in the original equation to avoid extraneous solutions ### Example 1: Solve $\sqrt{x} = 5$ Step 1: Square both sides $(\sqrt{x})^2 = 5^2 \Rightarrow x = 25$ Check: $\sqrt{25} = 5$ ### Example 2: Solve $\sqrt{x + 3} = 4$ Step 1: Square both sides $(\sqrt{x + 3})^2 = 4^2 \Rightarrow x + 3 = 16$ Step 2: Solve $x = 16 - 3 = 13$ Check: $\sqrt{13 + 3} = \sqrt{16} = 4$ ### Example 3: Solve $2 + \sqrt{x} = 7$ Step 1: Isolate the radical $\sqrt{x} = 7 - 2 = 5$ Step 2: Square both sides $x = 25$ Check: $2 + \sqrt{25} = 2 + 5 = 7$ ### Example 4: Solve an Equation from an Electrical Formula Suppose an AC voltage is calculated using: $V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}}$ Given: $V_{\text{RMS}} = 120$ volts Step 1: Multiply both sides by $\sqrt{2}$ $V_{\text{peak}} = 120 \cdot \sqrt{2}$ Step 2: Estimate $V_{\text{peak}} \approx 120 \cdot 1.414 = 169.7\ \text{V}$ ### Potential Pitfall: Extraneous Solutions Always check your solution in the original equation. Squaring both sides of an equation can sometimes introduce a solution that does not actually work in the original equation. ### Example 5: Check for Extraneous Solution Solve: $\sqrt{x} = -3$ No solution, because a square root cannot equal a negative number in the real number system. ### Summary - Isolate the radical first - Square both sides carefully - Solve the resulting equation - Always check your answer in the original