4x^2 - 78x + 360 = 252

["# How to Solve the Quadratic Equation 4x² – 78x + 360 = 252", "solving quadratic equations is a fundamental skill in algebra with wide-ranging applications in science, engineering, and economics. If you’re facing the equation 4x² – 78x + 360 = 252, simplifying and solving it step-by-step can turn confusion into clarity.", "## Understanding the Equation", "The given equation is:", "4x² – 78x + 360 = 252", "To solve for x, begin by simplifying both sides to bring the equation into standard quadratic form:", "4x² – 78x + 360 – 252 = 0\n4x² – 78x + 108 = 0", "Now the equation is:", "4x² – 78x + 108 = 0", "## Step 1: Simplify the Equation", "Before jumping into the quadratic formula, simplify coefficients to make calculations easier. Notice that all terms are divisible by 2:", "Divide every term by 2:", "2x² – 39x + 54 = 0", "This simplified form is easier to work with while preserving the solution accuracy.", "## Step 2: Identify Coefficients", "For the simplified standard quadratic equation ax² + bx + c = 0, we identify:", "- a = 2\n- b = –39\n- c = 54", "## Step 3: Apply the Quadratic Formula", "When factoring is difficult, the quadratic formula guarantees a solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 3.1: Calculate the Discriminant", "Compute the discriminant, D = b² – 4ac:", "[\nD = (-39)^2 - 4(2)(54) = 1521 - 432 = 1089\n]", "### Step 3.2: Take the Square Root of D", "[\n\sqrt{1089} = 33 \quad \ ext{(since 33 × 33 = 1089)}\n]", "### Step 3.3: Plug Values into the Formula", "[\nx = \frac{-(-39) \pm \sqrt{1089}}{2(2)} = \frac{39 \pm 33}{4}\n]", "### Step 3.4: Calculate Both Solutions", "[\nx_1 = \frac{39 + 33}{4} = \frac{72}{4} = 18\n]\n[\nx_2 = \frac{39 - 33}{4} = \frac{6}{4} = 1.5\n]", "## Final Answer", "The solutions to the equation 4x² – 78x + 360 = 252 are:", "- x = 18\n- x = 1.5", "## Why This Equation Matters", "Quadratic equations model real-world phenomena such as projectile motion, profit optimization, and physics problems. Mastering their solution helps unlock deeper insights and problem-solving skills.", "## Tips for Solving Similar Equations", "- Always simplify first: Look for common factors to reduce complexity.\n- Double-check discriminant: A positive discriminant indicates two real solutions; zero means one real solution, and negative implies complex ones.\n- Plug solutions back in: Validate results by substituting values into the original equation.", "---", "By following these clear steps, you can confidently solve quadratic equations like 4x² – 78x + 360 = 252, turning algebraic challenges into valuable mathematical insights."]