$$ x = \frac{4 \pm \sqrt{16 + 12}}{2} = \frac{4 \pm \sqrt{28}}{2} = 2 \pm \sqrt{7} $$
["Understanding the Equation ( x = \frac{4 \pm \sqrt{16 + 12}}{2} = 2 \pm \sqrt{7} ): A Simplified Explanation", "When solving quadratic equations, one common form is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), known as the quadratic formula. In this article, we will break down the equation ( x = \frac{4 \pm \sqrt{16 + 12}}{2} ) and simplify it to ( x = 2 \pm \sqrt{7} ), clarifying each step for better understanding.", "---", "### Deriving the Equation Step-by-Step", "Start with a general quadratic equation in the form:\n[\nax^2 + bx + c = 0\n]", "For our example, suppose ( a = 1 ), ( b = -4 ), and ( c = 3 ), giving:\n[\nx^2 - 4x + 3 = 0\n]", "Apply the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 1 ), ( b = -4 ), ( c = 3 ):\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} = \frac{4 \pm \sqrt{16 - 12}}{2}\n]", "Simplify the discriminant:\n[\n\sqrt{16 + 12} = \sqrt{28}\n]", "Now the expression becomes:\n[\nx = \frac{4 \pm \sqrt{28}}{2}\n]", "---", "### Simplifying the Square Root", "We simplify ( \sqrt{28} ) by factoring:\n[\n\sqrt{28} = \sqrt{4 \ imes 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}\n]", "So the expression becomes:\n[\nx = \frac{4 \pm 2\sqrt{7}}{2}\n]", "---", "### Final Simplification", "Factor and divide evenly:\n[\nx = \frac{4}{2} \pm \frac{2\sqrt{7}}{2} = 2 \pm \sqrt{7}\n]", "---", "### Why This Solution Matters", "This simplified form ( x = 2 \pm \sqrt{7} ) gives us two precise solutions:\n- ( x = 2 + \sqrt{7} )\n- ( x = 2 - \sqrt{7} )", "These values are crucial in algebra, calculus, and applied mathematics, particularly when modeling real-world phenomena such as projectile motion, optimization problems, or quadratic growth trends.", "---", "### Practical Applications & Key Takeaways", "- Quadratic equations appear frequently in physics and engineering for modeling curves and parabolic trajectories.\n- Simplifying radical expressions improves readability and usability in further calculations.\n- The ( \pm ) sign indicates both roots—critical for solving equations where multiple solutions exist.\n- Recognizing perfect squares under the root helps simplify expressions quickly.", "---", "In summary, converting ( x = \frac{4 \pm \sqrt{16 + 12}}{2} ) into ( x = 2 \pm \sqrt{7} ) is a fundamental algebraic skill that enhances clarity and application accuracy. Understanding this step reinforces core problem-solving abilities essential in advanced mathematics.", "---", "Keywords:\n$ x = \frac{4 \pm \sqrt{16 + 12}}{2} $, $ 2 \pm \sqrt{7} $, quadratic formula, simplifying radicals, algebra solutions, radical expressions, mathematical explanation", "Meta Description:\nExplore the step-by-step simplification of ( x = \frac{4 \pm \sqrt{16 + 12}}{2} ) into ( x = 2 \pm \sqrt{7} ). Learn key algebra techniques and applications in mathematics and science."]
