Squares Are Never Negative

The root of all inequalities

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $(a - b)^2 \ge 0$ gives $a^2 + b^2 \ge 2ab$.
• For $x > 0$: $x + \frac{k}{x} \ge 2\sqrt k$, with equality at $x = \sqrt k$.
• Equality cases tell you where the optimum is.

Steps

1. A simple start
2. Find the minimum
3. Your turn
4. Complete the square
5. Two variables
6. A fourth-power puzzle