A quadratic equation is given by \( ax^2 + bx + c = 0 \). If the roots are 3 and -5, find the equation given that \( a = 2 \).

["# Finding a Quadratic Equation with Roots 3 and -5 (Given ( a = 2 ))", "A quadratic equation has the standard form:\n[\nax^2 + bx + c = 0\n]\nWhen the roots of the equation are provided—here, 3 and -5—and the value of the leading coefficient ( a = 2 ), forming the equation becomes straightforward using the root-factor form.", "## Step 1: Use the Root-Factor Form\nIf ( x = 3 ) and ( x = -5 ) are the roots, the equation can be written as:\n[\na(x - 3)(x + 5) = 0\n]\nSince ( a = 2 ), substitute this value in:\n[\n2(x - 3)(x + 5) = 0\n]", "## Step 2: Expand the Factored Form\nFirst, expand the binomial expression:\n[\n(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15\n]\nNow multiply by the leading coefficient 2:\n[\n2(x^2 + 2x - 15) = 2x^2 + 4x - 30\n]", "## Step 3: Final Quadratic Equation\nThus, the quadratic equation with roots 3 and -5 and leading coefficient 2 is:\n[\n2x^2 + 4x - 30 = 0\n]", "---", "## Summary\nGiven roots of 3 and -5, and ( a = 2 ), the quadratic equation is:\n[\n\boxed{2x^2 + 4x - 30 = 0}\n]", "This form is useful in algebra, calculus, and physics, where modeling parabolic paths or solving quadratic relationships is required. Understanding how to build such equations from roots enables clear, algebra-based problem solving."]