Compute the sum of the roots of the equation \( u\sqrt{u} - 5u + 6\sqrt{u} = 0 \), given that all roots are non-negative.
["Title: How to Compute the Sum of the Roots for the Equation ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 )", "---", "When solving radical equations, especially those involving terms like ( u\sqrt{u} ) or ( \sqrt{u} ), understanding how to compute the sum of the roots is essential. In this article, we analyze the equation:", "[\nu\sqrt{u} - 5u + 6\sqrt{u} = 0\n]", "and guide you step-by-step to compute the sum of its non-negative real roots.", "---", "### Understanding the Equation", "The equation involves square roots and cubic-root-like terms (since ( u\sqrt{u} = u^{3/2} )), which are only defined for non-negative values of ( u ). So we restrict our attention to ( u \geq 0 ).", "Let’s simplify and solve this equation systematically.", "---", "### Step 1: Substitution to Eliminate the Radical", "To make the equation algebraically manageable, use the substitution:", "[\nx = \sqrt{u} \quad \Rightarrow \quad u = x^2, \quad u\sqrt{u} = x^2 \cdot x = x^3\n]", "Substituting into the original equation:", "[\nx^3 - 5x^2 + 6x = 0\n]", "This is now a cubic equation in ( x ), with ( x \geq 0 ) (since ( x = \sqrt{u} )).", "---", "### Step 2: Factor the Cubic Equation", "Factor out the common ( x ):", "[\nx(x^2 - 5x + 6) = 0\n]", "Now factor the quadratic:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "So the full factorization is:", "[\nx(x - 2)(x - 3) = 0\n]", "---", "### Step 3: Solve for ( x )", "Set each factor equal to zero:", "[\nx = 0, \quad x = 2, \quad x = 3\n]", "All solutions are non-negative, as required.", "---", "### Step 4: Convert Back to ( u )", "Recall ( u = x^2 ). Compute the corresponding ( u )-values:", "- For ( x = 0 ): ( u = 0^2 = 0 )\n- For ( x = 2 ): ( u = 2^2 = 4 )\n- For ( x = 3 ): ( u = 3^2 = 9 )", "Thus, the roots of the original equation are:", "[\nu = 0, \quad u = 4, \quad u = 9\n]", "---", "### Step 5: Compute the Sum of the Roots", "[\n0 + 4 + 9 = 13\n]", "---", "### Final Answer", "The sum of the roots of the equation ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 ) is:", "[\n\boxed{13}\n]", "---", "### Bonus TIPS for solving similar equations", "- Always substitute ( \sqrt{u} = x ) when dealing with terms like ( u\sqrt{u} ).\n- Factor carefully using algebraic identities.\n- Check domain restrictions (non-negative ( u )) due to radicals.\n- Convert back to original variable after solving to find valid roots.", "---", "Understanding how to transform and solve such equations helps in tackling advanced algebra and even problems in calculus and physics involving nonlinear models.", "---", "Keywords: compute sum of roots equation ( u\sqrt{u} - 5u + 6\sqrt{u} = 0 ), sum of roots formula, non-negative roots, substitution method, solving radical equations, algebra tutorial"]
