Here, \( a = 1, b = -4, c = -10 \)

["# Solving the Quadratic Equation: An In-Depth Look at ( a = 1, b = -4, c = -10 )", "When studying quadratic equations, understanding their structure and solutions is essential for both students and math enthusiasts. The equation defined by coefficients ( a = 1 ), ( b = -4 ), and ( c = -10 ) takes the standard form:", "[\nax^2 + bx + c = 0\n]", "Substituting the given values, we get:", "[\nx^2 - 4x - 10 = 0\n]", "## Why This Equation Matters", "Quadratic equations model many real-world phenomena, from projectile motion to optimization problems. Solving such equations helps reveal key properties like roots, symmetry, and vertex location—foundations in algebra and calculus.", "## Step-by-Step Solution Using the Quadratic Formula", "The quadratic formula provides a direct method to find the roots of any equation ( ax^2 + bx + c = 0 ):", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation ( x^2 - 4x - 10 = 0 ):", "- ( a = 1 )\n- ( b = -4 )\n- ( c = -10 )", "Now compute the discriminant ( D ), which determines the nature of the roots:", "[\nD = b^2 - 4ac = (-4)^2 - 4(1)(-10) = 16 + 40 = 56\n]", "Since ( D = 56 > 0 ), the equation has two distinct real roots.", "Next, calculate the square root of the discriminant:", "[\n\sqrt{56} = \sqrt{4 \ imes 14} = 2\sqrt{14}\n]", "Now substitute into the formula:", "[\nx = \frac{-(-4) \pm 2\sqrt{14}}{2(1)} = \frac{4 \pm 2\sqrt{14}}{2}\n]", "Simplify:", "[\nx = 2 \pm \sqrt{14}\n]", "## Final Roots", "Thus, the two real solutions are:", "[\nx_1 = 2 + \sqrt{14}, \quad x_2 = 2 - \sqrt{14}\n]", "Approximately, since ( \sqrt{14} \approx 3.7417 ), the roots are roughly:", "- ( x_1 \approx 5.7417 )\n- ( x_2 \approx -1.7417 )", "## Applications and Further Exploration", "This quadratic helps illustrate:", "- Vertex location: Using ( x = -b/(2a) ), the vertex is at ( x = 2 ).\n- Axis of symmetry: The line ( x = 2 ) bisects the parabola.\n- Graph shape: Opens upward since ( a = 1 > 0 ).\n- Root behavior: Since ( D > 0 ), roots lie on opposite sides of the vertex.", "## Practice and Motivation", "Understanding these equations builds a strong foundation for solving more complex problems in physics, economics, and engineering. Try substituting different coefficients to explore how changes affect the number and type of roots.", "---", "Keywords: quadratic equation ( x^2 - 4x - 10 = 0 ), roots, quadratic formula, discriminant, ( a = 1 ), ( b = -4 ), ( c = -10 ), ( 2 + \sqrt{14} ), ( 2 - \sqrt{14} )", "---", "Meta Description: Learn how to solve ( x^2 - 4x - 10 = 0 ) using the quadratic formula. Discover the roots, their significance, and real-world applications in algebra. Perfect for students and math learners."]