Question: Compute the square of the sum of the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \).
["# Compute the Square of the Sum of the Roots of the Quadratic Equation ( x^2 - 5x + 6 = 0 )", "When studying quadratic equations, one common question students encounter is how to compute key properties of the roots—such as their sum and product—without solving for the roots directly. In this article, we’ll focus on computing the square of the sum of the roots for the equation ( x^2 - 5x + 6 = 0 ), a straightforward yet fundamental problem in algebra.", "## Understanding the Quadratic Equation\nThe standard form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "For the equation ( x^2 - 5x + 6 = 0 ), we identify the coefficients:\n- ( a = 1 )\n- ( b = -5 )\n- ( c = 6 )", "## Using Vieta’s Formulas\nRather than factoring or using the quadratic formula, we apply Vieta’s formulas, which relate the coefficients of a quadratic equation to the sum and product of its roots.", "Let ( r_1 ) and ( r_2 ) be the roots of the equation. Vieta’s formulas tell us:\n- Sum of roots: ( r_1 + r_2 = -\frac{b}{a} )\n- Product of roots: ( r_1 \cdot r_2 = \frac{c}{a} )", "For our equation:\n[\nr_1 + r_2 = -\frac{-5}{1} = 5\n]", "## Compute the Square of the Sum\nNow, to find the square of the sum of the roots:", "[\n(r_1 + r_2)^2 = 5^2 = 25\n]", "Alternatively, using Vieta’s directly:", "[\n(r_1 + r_2)^2 = \left(-\frac{b}{a}\right)^2 = \left(-\frac{-5}{1}\right)^2 = 5^2 = 25\n]", "## Why This Matters\nComputing the square of the sum of the roots efficiently helps in solving problems involving symmetry, optimization, or geometry involving quadratic functions. It also highlights how Vieta’s formulas simplify complex algebraic operations using only coefficients.", "## Final Answer\n[\n\boxed{25}\n]", "This result shows that even without explicitly finding the roots, we can quickly determine ( (r_1 + r_2)^2 ) using the coefficients of the quadratic equation—making algebra both elegant and powerful."]
