u\sqrt{u} - 5u + 6\sqrt{u} = 0,
["Solving the Equation ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 ): A Clear Step-by-Step Guide", "When faced with transcendental equations involving both polynomial and radical terms, such as ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 ), breaking them into manageable parts and substituting variables can simplify the solution process. This SEO-optimized article provides a detailed, step-by-step explanation of how to solve this equation efficiently, making it valuable for students, educators, and math enthusiasts seeking clarity on radical and power equations.", "---", "### Understanding the Equation", "The equation at hand is:", "[\nu\sqrt{u} - 5u + 6\sqrt{u} = 0\n]", "Note that ( \sqrt{u} = u^{1/2} ), so we can rewrite every term involving square roots as fractional exponents. This facilitates easier manipulation and substitution.", "---", "### Step 1: Simplify Using Substitution", "To eliminate radicals and polynomial terms, let’s use the substitution:", "[\nx = \sqrt{u} \quad \Rightarrow \quad u = x^2\n]", "Then:\n- ( u\sqrt{u} = x^2 \cdot x = x^3 )\n- ( u = x^2 )\n- ( \sqrt{u} = x )", "Substituting into the original equation:", "[\nx^3 - 5x^2 + 6x = 0\n]", "---", "### Step 2: Factor the Polynomial", "We now solve:", "[\nx^3 - 5x^2 + 6x = 0\n]", "Factor out ( x ):", "[\nx(x^2 - 5x + 6) = 0\n]", "Next, factor the quadratic:", "[\nx(x - 2)(x - 3) = 0\n]", "---", "### Step 3: Solve for ( x )", "Set each factor equal to zero:", "- ( x = 0 )\n- ( x - 2 = 0 \Rightarrow x = 2 )\n- ( x - 3 = 0 \Rightarrow x = 3 )", "Thus, the solutions in variable ( x ) are ( x = 0, 2, 3 )", "---", "### Step 4: Back-Substitute to Find ( u )", "Recall that ( u = x^2 ), so:", "- For ( x = 0 ): ( u = 0^2 = 0 )\n- For ( x = 2 ): ( u = 2^2 = 4 )\n- For ( x = 3 ): ( u = 3^2 = 9 )", "---", "### Step 5: Verify Solutions in Original Equation", "We check each value to ensure no extraneous solutions were introduced.", "- ( u = 0 ): ( 0\cdot 0 - 5(0) + 6\cdot 0 = 0 \quad \checkmark )\n- ( u = 4 ): ( 4\cdot 2 - 5(4) + 6\cdot 2 = 8 - 20 + 12 = 0 \quad \checkmark )\n- ( u = 9 ): ( 9\cdot 3 - 5(9) + 6\cdot 3 = 27 - 45 + 18 = 0 \quad \checkmark )", "All solutions satisfy the original equation.", "---", "### Final Answer", "The solutions to the equation ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 ) are:", "[\n\boxed{u = 0,\ 4,\ 9}\n]", "---", "### Why This Method Works", "Using substitution to eliminate square roots transforms a mixed transcendental-polynomial equation into a standard polynomial form. Factoring and solving polynomial equations is a core algebraic technique, and matching back to the original variable ensures accuracy. This approach is essential for similar equations involving ( \sqrt{u} ) or ( u^{1/2} ).", "---", "### SEO Keywords", "- Solve ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 )\n- How to solve radical equations\n- Step-by-step radical equation solver\n- Equation with ( u\sqrt{u} )\n- Transcendental equation solution guide\n- Algebraic substitution method for radicals", "---", "### Additional Resources", "- How to Solve Equations with Square Roots\n- Polynomial Factorization Techniques\n- Radical Equations: Substitution Methods", "---", "Start mastering radical equations today with confidence—simplify, substitute, solve!"]
