["Understanding the Quadratic Equation with Coefficients ( a = 2 ), ( b = -3 ), ( c = -5 )", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), identifying the coefficients is crucial for applying the quadratic formula correctly. In this article, we explore the specific quadratic equation where ( a = 2 ), ( b = -3 ), and ( c = -5 ). This foundational understanding helps in finding its roots and analyzing its graph.", "### The Quadratic Equation", "Given:
\n( a = 2 ),
\n( b = -3 ),
\n( c = -5 ),", "the equation becomes:
\n[
\n2x^2 - 3x - 5 = 0
\n]", "This standard form allows direct use of the quadratic formula:
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "### Step-by-Step Calculation", "1. Identify coefficients:
\n ( a = 2 ), ( b = -3 ), ( c = -5 )", "2. Calculate the discriminant:
\n [
\n \Delta = b^2 - 4ac = (-3)^2 - 4(2)(-5) = 9 + 40 = 49
\n ]
\n Since ( \Delta > 0 ), two distinct real roots exist.", "3. Apply the quadratic formula:
\n [
\n x = \frac{-(-3) \pm \sqrt{49}}{2(2)} = \frac{3 \pm 7}{4}
\n ]", "4. Compute the two solutions:
\n [
\n x_1 = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2}
\n ]
\n [
\n x_2 = \frac{3 - 7}{4} = \frac{-4}{4} = -1
\n ]", "### Summary of Solutions
\nThe roots of the equation ( 2x^2 - 3x - 5 = 0 ) are:
\n- ( x = \frac{5}{2} )
\n- ( x = -1 )", "### Graphical Representation", "Plotting this quadratic function ( y = 2x^2 - 3x - 5 ) yields a parabola opening upwards (since ( a = 2 > 0 )), with its vertex positioned between the two roots. The x-intercepts at ( x = -1 ) and ( x = \frac{5}{2} ) are key points lying on the curve where it crosses the x-axis.", "### Real-World Applications", "Quadratic equations like this arise in engineering, physics, and economics—such as modeling projectiles, optimizing area given fixed perimeter, or analyzing profit functions where nonlinear relationships dominate.", "### Conclusion", "Using ( a = 2 ), ( b = -3 ), and ( c = -5 ), the quadratic equation becomes a clear example of applying the quadratic formula. Its discriminant confirms real roots, and solving step-by-step illustrates efficient root computation. Understanding this core equation enhances comprehension of quadratic behaviors essential across many scientific and mathematical domains.", "---", "Keywords: quadratic equation, roots of (2x^2 - 3x - 5 = 0), discriminant, quadratic formula, real solutions, parabola graphing."]
