Let $ t = rac{x^2 + y^2}{xy} $. By the AM-GM inequality,
["Understanding the Expression $ t = \dfrac{x^2 + y^2}{xy} $ Using the AM-GM Inequality", "The mathematical expression $ t = \dfrac{x^2 + y^2}{xy} $ arises often in algebra, optimization, and applied mathematics. Its elegant properties make it a great candidate for analysis using one of the most powerful inequalities in mathematics—the Arithmetic Mean–Geometric Mean (AM-GM) Inequality.", "---", "## What is AM-GM?", "The AM-GM inequality states that for any two positive real numbers $ x $ and $ y $,\n$$\n\frac{x + y}{2} \geq \sqrt{xy}\n$$\nEquality holds if and only if $ x = y $.", "While AM-GM traditionally applies to sums, it can be extended creatively to rational expressions such as $ \dfrac{x^2 + y^2}{xy} $.", "---", "## Analyzing the Given Expression", "We begin with:\n$$\nt = \frac{x^2 + y^2}{xy}\n$$", "Note that $ x $ and $ y $ must be positive real numbers (to keep the denominator non-zero and the logarithmic interpretation meaningful in many contexts).", "---", "### Step 1: Rewrite the Expression", "Split the fraction:\n$$\nt = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x}\n$$", "Now we have:\n$$\nt = \frac{x}{y} + \frac{y}{x}\n$$", "This form is more familiar and aligns well with AM-GM.", "---", "### Step 2: Apply AM-GM to the Terms", "Let $ a = \frac{x}{y} $, $ b = \frac{y}{x} $. Note that $ ab = 1 $.", "Then:\n$$\nt = a + b \geq 2\sqrt{ab} = 2\sqrt{1} = 2\n$$", "Thus, by AM-GM:\n$$\nt \geq 2\n$$", "Equality occurs when $ a = b $, i.e., $ \frac{x}{y} = \frac{y}{x} \Rightarrow x = y $.", "---", "### Step 3: Interpret the Result", "The inequality $ t \geq 2 $ reveals that the smallest possible value of $ t $ is 2, achieved precisely when $ x = y $. This sharp bound is highly valuable when minimizing or optimizing expressions involving $ x^2 + y^2 $ and $ xy $.", "---", "## Practical Implications", "- Optimization problems: When minimizing $ \frac{x^2 + y^2}{xy} $, setting $ x = y $ yields the minimum value of 2.\n- Inequalities: This identity is foundational in proving deeper inequalities in algebra and calculus.\n- Application domains: Used in engineering, economics, and physics where energy or efficiency ratios reduce to such forms.", "---", "## Conclusion", "By transforming the expression $ t = \dfrac{x^2 + y^2}{xy} $ into $ t = \frac{x}{y} + \frac{y}{x} $ and applying the AM-GM inequality, we elegantly prove:", "$$\n\boxed{t \geq 2}\n$$", "with equality if and only if $ x = y $. This insight not only simplifies analysis but also highlights the power of classical inequalities in modern mathematics.", "---", "Keywords: $ t = \dfrac{x^2 + y^2}{xy} $, AM-GM inequality, $ \frac{x}{y} + \frac{y}{x} \geq 2 $, algebraic inequality, optimization, real numbers, positive real numbers, $ x = y $."]
