A quadratic equation \(x^2 - 5x + 6 = 0\) has roots \(r_1\) and \(r_2\). What is the value of \(r_1^2 + r_2^2\)?

["Understanding (r_1^2 + r_2^2) for the Quadratic Equation (x^2 - 5x + 6 = 0)", "Solving quadratic equations is a fundamental skill in algebra, and one useful concept is calculating the sum of the squares of its roots. For the quadratic equation (x^2 - 5x + 6 = 0), identifying the roots and computing (r_1^2 + r_2^2) offers both theoretical insight and practical value.", "---", "### What Are the Roots (r_1) and (r_2)?", "The given quadratic equation is:\n[\nx^2 - 5x + 6 = 0\n]", "Factoring this equation is straightforward because we look for two numbers that multiply to (6) (the constant term) and add to (-5) (the coefficient of (x)). These numbers are (-2) and (-3), so:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Thus, the roots are:\n[\nr_1 = 2 \quad \ ext{and} \quad r_2 = 3\n]", "---", "### Why Calculate (r_1^2 + r_2^2)?", "Calculating (r_1^2 + r_2^2) helps verify properties of the roots without explicitly solving for them each time. This value appears in many mathematical contexts, including geometry, infinite series, and error analysis.", "---", "### Efficient Calculation Using Algebraic Identities", "Instead of squaring and adding the roots separately, we use an elegant identity involving the sum and product of roots:", "[\nr_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2\n]", "This formula comes directly from expanding ((r_1 + r_2)^2 = r_1^2 + 2r_1 r_2 + r_2^2), then subtracting (2r_1 r_2) to isolate (r_1^2 + r_2^2).", "---", "### Apply Vieta’s Formulas", "For a general quadratic equation (ax^2 + bx + c = 0), Vieta’s formulas tell us:", "[\nr_1 + r_2 = -\frac{b}{a}, \quad r_1 r_2 = \frac{c}{a}\n]", "In our equation, (a = 1), (b = -5), (c = 6):", "[\nr_1 + r_2 = -\frac{-5}{1} = 5\n]\n[\nr_1 r_2 = \frac{6}{1} = 6\n]", "Now substitute into the identity:", "[\nr_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = 5^2 - 2 \ imes 6 = 25 - 12 = 13\n]", "---", "### Final Answer", "[\n\boxed{13}\n]", "So, the value of (r_1^2 + r_2^2) for the equation (x^2 - 5x + 6 = 0) is (13), computable efficiently using algebraic identities and Vieta’s formulas.", "---", "### Why This Matters", "Understanding how to compute expressions like (r_1^2 + r_2^2) strengthens foundational algebra skills. These techniques apply across higher mathematics, physics, engineering, and data science, where analyzing roots and their relationships is essential.", "For anyone learning quadratics, mastering this shortcut saves time and deepens conceptual insight—making (x^2 - 5x + 6 = 0) not just a problem to solve, but a tool that teaches timeless mathematical reasoning."]