\[x^2 + 25x - 48 = 0\]

["# Solving the Quadratic Equation: (x^2 + 25x - 48 = 0)", "Solving quadratic equations is a fundamental skill in algebra that appears in various real-world applications—from physics and engineering to economics. One such equation widely used in educational curricula is:", "[\nx^2 + 25x - 48 = 0\n]", "This article will guide you through understanding, solving, and interpreting the solutions of this quadratic equation using both analytical methods and practical insights.", "---", "## What Is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[\nax^2 + bx + c = 0\n]", "where (a), (b), and (c) are constants, and (a <br/>\ne 0). The solutions, or roots, represent the values of (x) that satisfy the equation. These roots can be found using:", "- The quadratic formula\n- Factoring\n- Completing the square\n- Graphical analysis", "---", "## Step 1: Identify Coefficients", "For the equation\n[\nx^2 + 25x - 48 = 0\n]\nwe identify:\n- (a = 1)\n- (b = 25)\n- (c = -48)", "---", "## Step 2: Apply the Quadratic Formula", "The most reliable method for solving any quadratic equation is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substituting the values:", "[\nx = \frac{-25 \pm \sqrt{25^2 - 4(1)(-48)}}{2(1)}\n]", "Calculate the discriminant ((D)):", "[\nD = b^2 - 4ac = 625 + 192 = 817\n]", "Since (D > 0), there are two distinct real roots.", "---", "## Step 3: Compute the Roots", "Now compute:", "[\nx = \frac{-25 \pm \sqrt{817}}{2}\n]", "The square root of 817 is irrational (approximately ( \sqrt{817} \approx 28.58 )), so we express the exact solutions as:", "[\nx = \frac{-25 + \sqrt{817}}{2} \quad \ ext{and} \quad x = \frac{-25 - \sqrt{817}}{2}\n]", "These are the precise algebraic solutions.", "---", "## Step 4: Approximate the Solutions", "For practical purposes, approximate decimals are useful:", "[\nx \approx \frac{-25 + 28.58}{2} \approx 1.79\n]", "[\nx \approx \frac{-25 - 28.58}{2} \approx -26.79\n]", "Thus, the equation (x^2 + 25x - 48 = 0) has two real roots: approximately (1.79) and (-26.79).", "---", "## Step 5: Verify the Solutions", "It’s always good practice to check the roots by plugging them back into the original equation.", "For (x \approx 1.79):", "[\n(1.79)^2 + 25(1.79) - 48 \approx 3.2 + 44.75 - 48 \approx 0 \quad \ ext{(close enough)}\n]", "For (x \approx -26.79):", "[\n(-26.79)^2 + 25(-26.79) - 48 \approx 718.5 - 669.75 - 48 \approx -49.25 + 718.5 - 48 \approx 0 \quad \ ext{(within rounding error)}\n]", "---", "## Step 6: Graphical Interpretation", "Plotting the quadratic function (f(x) = x^2 + 25x - 48) yields a parabola opening upwards (since (a = 1 > 0)). The roots are the x-intercepts where the function crosses the x-axis—located near (x \approx 1.79) and (x \approx -26.79), matching our analytical solutions.", "---", "## Why This Equation Matters", "Quadratic equations model phenomena such as projectile motion, profit maximization, and optimization problems. Understanding how to solve and interpret these equations equips learners with powerful tools for academic and real-life problem-solving.", "---", "## Summary", "- The equation (x^2 + 25x - 48 = 0) has two real roots.\n- Solving via the quadratic formula yields exact solutions:\n [\n x = \frac{-25 \pm \sqrt{817}}{2}\n ]\n- Approximate values are (x \approx 1.79) and (x \approx -26.79).\n- These roots represent the x-coordinates where the graph intersects the x-axis.\n- Useful in algebra, physics, economics, and engineering.", "---", "## Further Reading", "- Quadratic Formula Derivation\n- Graphing Quadratics and Their Roots\n- Discriminant Analysis of Root Nature", "---", "Keywords: solve quadratic equation (x^2 + 25x - 48 = 0), quadratic formula, real roots, discriminant, algebraic solutions, graphing quadratics, intermediate algebra."]