A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and -5. Find the values of \( a \), \( b \), and \( c \) if \( a = 2 \).
["Quadratic Equation Roots 3 and -5: Find a, b, and c When a = 2", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), knowing the roots can help easily determine the coefficients. In this article, we’ll explore how to find the values of ( a ), ( b ), and ( c ) given that the roots are ( x = 3 ) and ( x = -5 ), and specifically, when the leading coefficient ( a = 2 ).", "---", "### Understanding Roots and Their Relationship to Coefficients", "The roots of a quadratic equation represent the values of ( x ) that satisfy ( ax^2 + bx + c = 0 ). For a quadratic equation with roots ( r_1 ) and ( r_2 ), the equation can be expressed in factored form as:", "[\na(x - r_1)(x - r_2) = 0\n]", "Expanding this gives:", "[\na(x - 3)(x + 5) = 0\n]", "Because ( a = 2 ), substitute that in:", "[\n2(x - 3)(x + 5) = 0\n]", "---", "### Step 1: Expand the Factored Form", "Multiply the factor pairs:", "[\n(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15\n]", "Now multiply by ( a = 2 ):", "[\n2(x^2 + 2x - 15) = 2x^2 + 4x - 30\n]", "---", "### Step 2: Identify Coefficients ( a ), ( b ), and ( c )", "Comparing ( 2x^2 + 4x - 30 = 0 ) to the standard form ( ax^2 + bx + c = 0 ), we find:", "- ( a = 2 )\n- ( b = 4 )\n- ( c = -30 )", "---", "### Summary", "Given the quadratic equation ( ax^2 + bx + c = 0 ) has roots 3 and -5, and ( a = 2 ), the coefficients are:", "[\na = 2,\quad b = 4,\quad c = -30\n]", "This method ensures a quick derivation of the quadratic equation from its roots and the given leading coefficient.", "---", "### Why This Matters", "Understanding how roots determine coefficients is essential in algebra, algebra problems, and applications in physics and engineering. Using ( a = 2 ) here ties into scaling the basic equation with real-world scenarios or graphing needs.", "---", "Keywords: quadratic equation roots 3 and -5, find a b c when a = 2, solve quadratic with given roots, a(2)x² + bx + c = 0, expand factored form, coefficient calculation."]
