How to Solve Radical Equations and Check for Extraneous Solutions

The Organic Chemistry Tutor

This comprehensive mathematics tutorial guides students through the process of solving radical equations, progressing from simple examples to complex problems involving multiple radicals and quadratic steps. The video systematically demonstrates the essential algorithm for these problems: isolating the radical, raising both sides to the appropriate power to eliminate the root, solving the resulting linear or quadratic equation, and critically, checking for extraneous solutions. The lesson explores a wide variety of equation types, including those with square roots, cube roots, fractional exponents, and variables on both sides of the equation. A significant portion of the video is dedicated to a complex "challenge problem" containing two radical terms, which requires squaring both sides twice and using advanced factoring techniques. The narrator emphasizes procedural fluency and the logic behind each step, particularly why squaring algebraic expressions can introduce false solutions. For educators, this video serves as an excellent instructional tool for Algebra II or Pre-Calculus units on radical functions. It provides clear, step-by-step visual modeling of algebraic manipulation that students can follow along with. The progression of difficulty allows teachers to use the first half for basic instruction and the latter half for extension or advanced practice, specifically highlighting common pitfalls like failing to FOIL correctly when squaring binomials containing radicals.