x = \frac{4 \pm \sqrt{16 + 48}}{4}

["# Solving the Quadratic Equation: A Step-by-Step Guide to ( x = \frac{4 \pm \sqrt{16 + 48}}{4} )", "Quadratic equations are fundamental in algebra and appear in many real-world applications, from physics to engineering. One commonly encountered form is the standard quadratic solution:\n[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]\nBut sometimes, expressions under the square root seem puzzling—for example, ( x = \frac{4 \pm \sqrt{16 + 48}}{4} ). This article breaks down how to interpret and simplify this equation, solving it step by step while exploring its mathematical meaning and relevance.", "---", "## Understanding the Full Equation", "The given expression is a solution form derived from solving a quadratic equation of the form:\n[ ax^2 + bx + c = 0 ]\nFor your specific case:\n[ x = \frac{4 \pm \sqrt{16 + 48}}{4} ]", "This suggests the original quadratic equation had coefficients ( a = 4 ), ( b = -4 ), and ( c = 48 ), because:\n- The numerator (4 \pm \sqrt{16 + 48}) matches the discriminant-based solution\n- The denominator equals ( 2a = 2 \ imes 4 = 4 )", "So the full quadratic equation is:\n[ 4x^2 - 4x + 48 = 0 ]", "---", "## Step 1: Simplify the Discriminant", "Start with the discriminant ( D = b^2 - 4ac ):\n[\nD = (-4)^2 - 4(4)(48) = 16 - 768 = -752\n]", "Since the discriminant is negative (( D = -752 )), the equation has two complex roots, not real solutions.", "---", "## Step 2: Express the Full Solution", "Now substitute into the quadratic formula:\n[\nx = \frac{4 \pm \sqrt{-752}}{4}\n]\nBecause ( \sqrt{-752} = \sqrt{-1 \cdot 752} = i\sqrt{752} ), we rewrite:\n[\nx = \frac{4 \pm i\sqrt{752}}{4}\n]", "---", "## Step 3: Simplify the Square Root", "Simplify ( \sqrt{752} ):\n[\n752 = 16 \ imes 47 \Rightarrow \sqrt{752} = \sqrt{16 \ imes 47} = 4\sqrt{47}\n]\nThus,\n[\nx = \frac{4 \pm i \cdot 4\sqrt{47}}{4} = \frac{4(1 \pm i\sqrt{47})}{4} = 1 \pm i\sqrt{47}\n]", "---", "## Step 4: Final Solutions", "The two complex solutions are:\n[\nx = 1 + i\sqrt{47} \quad \ ext{and} \quad x = 1 - i\sqrt{47}\n]", "These represent the complex roots of the equation ( 4x^2 - 4x + 48 = 0 ), demonstrating how algebra extends beyond real numbers to include imaginary solutions.", "---", "## Why This Matters: Applications and Insights", "Solving equations with complex results may seem abstract, but they are critical in:\n- Electrical engineering (analyzing AC circuits with impedance)\n- Signal processing (Fourier transforms and filters)\n- Quantum mechanics (wave functions in complex probability amplitudes)", "Understanding how to manipulate expressions like ( x = \frac{4 \pm \sqrt{16 + 48}}{4} ) builds the foundational skill to tackle advanced equations involving complex numbers and higher-order polynomials.", "---", "## Summary", "- The original expression arises from the quadratic formula when ( a = 4 ), ( b = -4 ), ( c = 48 )\n- The discriminant is negative, indicating complex roots\n- Simplification leads to ( x = 1 \pm i\sqrt{47} )\n- Complex solutions extend algebra into vital areas of science and technology", "---", "###Keywords for SEO optimization:\nquadratic equation solution, complex roots of quadratic, solving \( x = \frac{4 \pm \sqrt{16 + 48}}{4} \), imaginary numbers in algebra, quadratic formula application, complex number arithmetic, discriminant analysis", "---", "Whether you're solving equations in class or exploring application domains, mastering these steps ensures a strong grasp of quadratic solutions—even when results lie beyond the real number line."]