Solution: Let us make the substitution $ x = \sqrt{u} $, so that $ u = x^2 $. Substituting into the equation gives:

["Title: The Power of Substitution in Solving Equations: Letting $ x = \sqrt{u} $ to Simplify Complex Expressions", "When faced with challenging mathematical equations, especially those involving radicals or fractional exponents, substitution often becomes the key to simplification and eventual solution. One particularly effective technique is letting $ x = \sqrt{u} $, which transforms the original equation into a more manageable form—often polynomial—making analysis, differentiation, integration, or numerical solving significantly easier.", "### Why Substitution Matters", "In many algebraic and calculus-based problems, expressions with roots or inverses complicate direct solving. Substitutions allow us to reframe these problems into simpler terms. By substituting $ x = \sqrt{u} $, we convert $ u $ into $ x^2 $, eliminating the square root and enabling polynomial manipulation. This transformation is especially valuable in contexts such as integration, differential equations, and optimization.", "---", "### The Substitution: $ x = \sqrt{u} $", "Let’s examine the substitution in detail. Assume our original equation contains $ u = x^2 $, meaning:", "$$\nx = \sqrt{u}\n$$", "This is a direct inverse relationship enabling bidirectional transformation:", "- From $ u $ to $ x $: $ x = \sqrt{u} \implies u = x^2 $\n- From $ x $ to $ u $: $ u = x^2 $", "Suppose we have an equation involving $ u $ and $ x $, such as:", "$$\nf(u) + g(x) = 0\n$$", "Substituting $ u = x^2 $, the equation becomes:", "$$\nf(x^2) + g(x) = 0\n$$", "This transformed form often reduces complexity, especially when $ f(x^2) $ turns into a polynomial in $ x^2 $, allowing standard algebraic or symbolic solution tools to apply.", "---", "### Practical Advantages", "- Eliminates Radicals: Converting $ \sqrt{u} $ into $ x $ removes irrational expressions, simplifying differentiation and integration.\n- Polynomial Eases Computation: Polynomials in $ x $ are easier to solve, factor, or apply numerical methods.\n- Improves Readability: The substitution clarifies relationships and supports structured problem-solving workflows.\n- Enables Advanced Techniques: Once reduced, methods like series expansions, substitution in integrals, or implicit differentiation become readily accessible.", "---", "### Example: Simplifying a Radical Equation", "Consider the equation:", "$$\n\sqrt{u} + 3u = 10\n$$", "Let $ x = \sqrt{u} $, so $ u = x^2 $. Substituting yields:", "$$\nx + 3x^2 = 10\n$$", "Rewriting:", "$$\n3x^2 + x - 10 = 0\n$$", "This quadratic is easily solvable using the quadratic formula:", "$$\nx = \frac{-1 \pm \sqrt{1 + 120}}{6} = \frac{-1 \pm \sqrt{121}}{6} = \frac{-1 \pm 11}{6}\n$$", "Solutions: $ x = \frac{10}{6} = \frac{5}{3} $ or $ x = -2 $", "Since $ x = \sqrt{u} \geq 0 $, we reject $ x = -2 $. Thus, $ \sqrt{u} = \frac{5}{3} \Rightarrow u = \left( \frac{5}{3} \right)^2 = \frac{25}{9} $", "This illustrates how substitution turns a radical equation into a clean quadratic.", "---", "### When to Use $ x = \sqrt{u} $", "This substitution is ideal when:", "- The equation includes $ \sqrt{u} $ or similar radical terms.\n- The equation involves $ u $ in a composite form that complicates differentiation or integration.\n- Solving for $ u $ explicitly requires eliminating the square root.", "---", "### Conclusion", "The substitution $ x = \sqrt{u} $ is a powerful algebraic tool that simplifies equations dominated by square roots. By transforming $ u = x^2 $, complex expressions become polynomials, unlocking efficient solution paths across calculus, differential equations, and applied mathematics. Whether you're solving integrals, optimizing functions, or working with parametric models, mastering such substitutions enhances your problem-solving versatility and accuracy.", "Try substituting $ x = \sqrt{u} $ the next time you encounter a radical equation—transform complexity into clarity.", "---", "Keywords: substitution $ x = \sqrt{u} $, simplify radical equations, solve $ \sqrt{u} $, algebra transformation, calculus techniques, mathematical problem solving, equation substitution method."]