r = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3} = \frac{6 \pm \sqrt{36 + 528}}{6} = \frac{6 \pm \sqrt{564}}{6}

["Understanding the Quadratic Formula: Deriving and Simplifying $ r = \frac{6 \pm \sqrt{564}}{6} $", "When solving quadratic equations, the quadratic formula is one of the most powerful tools available to mathematicians, engineers, and scientists. One such expression commonly derived — and widely applicable in real-world problems — is:", "$$\nr = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3} = \frac{6 \pm \sqrt{564}}{6}\n$$", "But how is this formula derived, and why is simplifying the square root of 564 important in mathematical modeling and physics? Let’s explorie step-by-step.", "---", "### 1. The Quadratic Equation: From Standard Form to Solutions", "The general form of a quadratic equation is:\n$$\nax^2 + bx + c = 0\n$$", "For our expression, the coefficients are:\n- $ a = 3 $\n- $ b = 6 $\n- $ c = -44 $", "Using the quadratic formula:\n$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Plugging in the values:\n$$\nr = \frac{-6 \pm \sqrt{(6)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3}\n= \frac{-6 \pm \sqrt{36 + 528}}{6}\n= \frac{-6 \pm \sqrt{564}}{6}\n$$", "---", "### 2. Simplifying $ \sqrt{564} $: Why It Matters", "At first glance, $ \sqrt{564} $ seems complex. However, simplifying radical expressions enhances readability, facilitates approximation, and aids in analytical solutions.", "Let’s simplify $ \sqrt{564} $:", "Factor 564:\n$$\n564 = 4 \ imes 141 = 4 \ imes 3 \ imes 47\n$$", "Thus,\n$$\n\sqrt{564} = \sqrt{4 \ imes 141} = 2\sqrt{141}\n$$", "Substitute back into the formula:\n$$\nr = \frac{-6 \pm 2\sqrt{141}}{6}\n$$", "Factor numerator:\n$$\nr = \frac{2(-3 \pm \sqrt{141})}{6} = \frac{-3 \pm \sqrt{141}}{3}\n$$", "So the simplified form is:\n$$\nr = \frac{-3 \pm \sqrt{141}}{3}\n$$", "---", "### 3. Applications of Quadratic Solutions", "Equations of the form $ ax^2 + bx + c = 0 $ model numerous real-world scenarios, including:", "- Projectile Motion: Calculating time or height when thrown under gravity.\n- Optimization Problems: Finding maximum profit or minimum cost in economics.\n- Engineering Design: Stability analysis in structural design.\n- Physics and Chemistry: Equilibrium points, reaction rates, and resonance frequencies.", "Simplifying such roots allows faster computation and clearer physical interpretation.", "---", "### 4. Why This Formula Is Essential in STEM", "Mastering the derivation and simplification of quadratic formulas empowers learners and professionals alike:", "- Enables faster problem-solving in competitive fields.\n- Helps identify nature of solutions via discriminant analysis ($ b^2 - 4ac $).\n- Facilitates engineering approximations and modeling.", "---", "### Summary", "The quadratic formula application expressed as:\n$$\nr = \frac{6 \pm \sqrt{564}}{6} = \frac{-3 \pm \sqrt{141}}{3}\n$$\nis not just an algebraic exercise — it represents a gateway to solving real-world problems efficiently. Simplifying radicals like $ \sqrt{564} $ to $ 2\sqrt{141} $ streamlines computation and deepens conceptual understanding.", "Whether in physics, finance, or engineering, mastering these formulas is key to unlocking precise, reliable analysis.", "---", "Keywords for SEO:\nquadratic formula, solve quadratic equation, simplify square roots, $ r = \frac{6 \pm \sqrt{564}}{6} $, quadratic solutions, discriminant application, STEM math formulas, algebra derivation, simplify radicals, real-world problem solving", "Meta Description:\nDiscover how to derive and simplify $ r = \frac{6 \pm \sqrt{564}}{6} $ using the quadratic formula, learn its significance in physics and engineering, and simplify radicals for clearer, faster calculations.", "---", "Ready to master quadratics and elevate your math skills? Start applying this formula in your next problem today!"]