x = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4}

["# Solving Quadratic Equations: A Detailed Guide to ( x = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4} )", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, scientists, and anyone working with mathematical modeling. In this article, we break down the elegant solution to the quadratic equation expressed as:", "[\nx = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4}\n]", "We will simplify, solve, and apply this formula step-by-step—helping you understand not just the result, but how and why it works.", "---", "## Understanding the Quadratic Formula", "A general quadratic equation takes the form:", "[\nax^2 + bx + c = 0\n]", "The solutions are found using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In our case, comparing with the general form:", "- ( a = 2 )\n- ( b = 39 )\n- ( c = 54 )", "Plugging these into the formula naturally leads to:", "[\nx = \frac{-39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4}\n]", "But since the expression ( x = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4} ) appears, we note a subtle sign difference in the numerator’s sign—likely intentional and consistent with specific conventions or rewrites. We will explain both perspectives.", "---", "## Step-by-Step Simplification", "### Step 1: Compute the discriminant ( D = b^2 - 4ac )", "[\nD = 39^2 - 4 \cdot 2 \cdot 54\n]", "First calculate:", "[\n39^2 = 1521\n]", "[\n4 \cdot 2 \cdot 54 = 8 \cdot 54 = 432\n]", "[\nD = 1521 - 432 = 1089\n]", "### Step 2: Take the square root of the discriminant", "[\n\sqrt{D} = \sqrt{1089} = 33\n]", "Since ( 33^2 = 1089 ), this confirms the square root simplifies neatly.", "### Step 3: Substitute values into the formula", "Using the standard quadratic formula:", "[\nx = \frac{-39 \pm \sqrt{1089}}{2 \cdot 2} = \frac{-39 \pm 33}{4}\n]", "Now compute both solutions:", "- ( x_1 = \frac{-39 + 33}{4} = \frac{-6}{4} = -1.5 )\n- ( x_2 = \frac{-39 - 33}{4} = \frac{-72}{4} = -18 )", "---", "## How This Formula Works Intuitively", "The quadratic formula calculates the roots of any degree-2 polynomial by averaging the two solutions provided by the ± expression. The discriminant determines the nature of the roots:", "- If ( D > 0 ): two distinct real roots (here, ( D = 1089 ))\n- If ( D = 0 ): one repeated real root\n- If ( D < 0 ): complex roots", "In our case, a positive discriminant confirms two real roots, which match our computed values.", "---", "## Real-World Applications", "Quadratic equations model parabolic trajectories, physics problems (like projectile motion), optimization challenges, and financial calculations involving growth models. Understanding the exact solution using the quadratic formula empowers you to analyze systems that behave quadratically.", "---", "## Final Answer", "[\nx = \frac{-39 \pm 33}{4}\n]", "Which yields the exact solutions:", "[\nx_1 = -1.5 \quad \ ext{or} \quad x_2 = -18\n]", "---", "## Summary", "Working through ( x = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4} ) reveals not just a numerical result, but a robust technique for solving quadratic equations. By computing the discriminant, applying square roots, and simplifying using the quadratic formula, we uncover two real solutions with clarity and precision.", "Whether you're a student mastering algebra or a professional applying math to real-world problems, mastering this process strengthens your analytical foundation.", "---", "Keywords: quadratic formula, solve quadratic equation, discriminant, ( x = \frac{39 \pm \sqrt{39^2 - 4 \cdot 2 \cdot 54}}{4} ), solve square roots, algebra tutorial, root calculation, math formula breakdown.", "---", "Related reading:\n- Step-by-step guide to completing the square\n- How to analyze quadratic functions\n- Quadratic equations: word problems and applications"]