\[ x = \frac{-2 \pm \sqrt{4 + 540}}{2} = \frac{-2 \pm \sqrt{544}}{2} \]

["# Solving the Quadratic Equation: ( x = \frac{-2 \pm \sqrt{544}}{2} ) Explained", "When solving quadratic equations, understanding how to simplify radical expressions is key to finding exact solutions. This article explores the step-by-step derivation and solution of the equation:", "[\nx = \frac{-2 \pm \sqrt{4 + 540}}{2} = \frac{-2 \pm \sqrt{544}}{2}\n]", "## Step 1: Simplify the Expression Under the Square Root", "First, simplify the expression inside the square root:", "[\n\sqrt{4 + 540} = \sqrt{544}\n]", "Next, simplify ( \sqrt{544} ). Begin by factoring 544 to identify perfect square factors:", "- ( 544 = 16 \ imes 34 )\n- Since ( 16 = 4^2 ), we write:", "[\n\sqrt{544} = \sqrt{16 \ imes 34} = \sqrt{16} \cdot \sqrt{34} = 4\sqrt{34}\n]", "Substituting this back, the equation becomes:", "[\nx = \frac{-2 \pm 4\sqrt{34}}{2}\n]", "## Step 2: Simplify the Fraction", "Now simplify each term by dividing numerator terms by the denominator:", "[\nx = \frac{-2}{2} \pm \frac{4\sqrt{34}}{2} = -1 \pm 2\sqrt{34}\n]", "So, the simplified exact solutions are:", "[\nx = -1 + 2\sqrt{34} \quad \ ext{and} \quad x = -1 - 2\sqrt{34}\n]", "## Step 3: Why This Equation Matters", "Quadratic equations of the form ( ax^2 + bx + c = 0 ) frequently appear in algebra, physics, engineering, and economics. Solving them gives the x-intercepts of corresponding parabolas — useful in modeling parabolic motion, optimization problems, and data analysis.", "Using exact radical form avoids rounding errors and ensures accuracy, especially in contexts requiring precision such as scientific calculations or advanced mathematics.", "### Final Solutions:", "[\nx = -1 + 2\sqrt{34} \quad \ ext{and} \quad x = -1 - 2\sqrt{34}\n]", "---", "Key SEO Keywords:\n( x = \frac{-2 \pm \sqrt{544}}{2} ), simplify ( \sqrt{544} ), quadratic equation solutions, exact radical form, solving quadratics algebraically, simplify square roots, algebra simplification tips, exact solutions for quadratics.", "---", "Summary:\nThe expression ( x = \frac{-2 \pm \sqrt{544}}{2} ) simplifies elegantly to ( x = -1 \pm 2\sqrt{34} ), revealing precise real roots without approximation. Understanding this process strengthens foundational algebra skills vital for STEM fields and analytical problem solving."]