Solving x² = 2ˣ : When Algebra Isn't Enough!

Most people look at x² = 2ˣ and confidently say it has two solutions. They're wrong — there are three. And the third one is hiding where almost nobody thinks to look. In this video, we start with the two obvious solutions (x = 2 and x = 4), then use graphical intuition to reveal why a third real solution must exist on the negative side of the number line. From there, we apply the Newton–Raphson method to approximate it precisely: x ≈ −0.7666. But there's a twist. A popular logarithmic approach to this equation finds only two solutions — and the reason it misses the third is a subtle, important mathematical lesson about how algebraic manipulations can silently restrict the domain of a problem. This video covers: ✔ Verifying x = 2 and x = 4 as exact solutions ✔ Why the rough graphical sketch of y = x² and y = 2ˣ is commonly drawn wrong ✔ How to detect and locate the hidden negative solution geometrically ✔ The Newton–Raphson iteration method applied step by step ✔ Why the logarithmic method misses the third root — and what that teaches us Whether you're a high school student, a math olympiad aspirant, or just someone who loves elegant problems, this video shows how the most interesting mathematics often begins with a question that seems too simple to take seriously. 🔔 Subscribe for more problems that reward curiosity over speed. #Mathematics #Algebra #Equations #NewtonRaphson #MathOlympiad #Calculus #MathPuzzle #ExponentialEquations