A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and -5. If \( a = 2 \), what is the value of \( b \)?
["Understanding the Quadratic Equation with Roots 3 and -5 — Finding ( b ) When ( a = 2 )", "When tasked with solving quadratic equations or working with polynomial roots, understanding how the coefficients relate to the roots is essential. In this article, we explore a key quadratic equation ( ax^2 + bx + c = 0 ) that has roots ( x = 3 ) and ( x = -5 ), with a given coefficient ( a = 2 ). We’ll use the relationship between roots and coefficients to find ( b ), and explain the underlying math for clarity.", "---", "### What Determines the Coefficients from the Roots?", "A quadratic equation can be expressed in factored form using its roots:", "[\na(x - r_1)(x - r_2) = 0\n]", "where ( r_1 ) and ( r_2 ) are the roots, and ( a ) is a leading coefficient. In this problem:\n- Roots: ( r_1 = 3 ), ( r_2 = -5 )\n- Given ( a = 2 )", "So the equation becomes:", "[\n2(x - 3)(x + 5) = 0\n]", "---", "### Step 1: Expand the Factored Form", "First, expand ( (x - 3)(x + 5) ):", "[\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]", "So the quadratic equation is:", "[\n2x^2 + 4x - 30 = 0\n]", "This matches the standard form ( ax^2 + bx + c = 0 ), where:\n- ( a = 2 )\n- ( b = 4 )\n- ( c = -30 )", "---", "### Step 2: Verify Using Sum and Product of Roots", "There’s a shortcut: for a quadratic ( ax^2 + bx + c = 0 ), if roots are ( p ) and ( q ), then:", "- Sum of roots: ( p + q = -\frac{b}{a} )\n- Product: ( p \cdot q = \frac{c}{a} )", "Given roots ( 3 ) and ( -5 ):", "Sum of roots:\n[\n3 + (-5) = -2 \quad \Rightarrow \quad -\frac{b}{a} = -2\n]", "Substitute ( a = 2 ):", "[\n-\frac{b}{2} = -2 \quad \Rightarrow \quad \frac{b}{2} = 2 \quad \Rightarrow \quad b = 4\n]", "Product of roots:\n[\n3 \cdot (-5) = -15 \quad \Rightarrow \quad \frac{c}{a} = -15\n]", "With ( a = 2 ), this gives ( c = -30 ), consistent with earlier.", "---", "### Conclusion: The Value of ( b )", "Using both expansion and the root-sum formula, we confirm that when ( a = 2 ) and the roots are ( 3 ) and ( -5 ), the coefficient ( b ) is:", "[\n\boxed{4}\n]", "---", "### Why This Matters", "This process is vital in algebra, mathematics education, and applications in physics, engineering, and computer science where modeling real-world phenomena with quadratic equations is common. Knowing how to derive coefficients from roots streamlines problem-solving and deepens conceptual understanding.", "For quick verification, always recall:\n[\nb = a \ imes (\ ext{sum of roots}) = 2 \ imes (3 - 5) = 2 \ imes (-2) = 4\n]", "So, the value of ( b ) is efficiently found without full expansion — ideal when working with known roots and leading coefficients.", "---", "Keywords: quadratic equation roots, find b in ax² + bx + c = 0, roots 3 and -5, verify using sum and product of roots, a = 2, b value, algebraic shortcuts, algebra tutorial.\nMeta description: Learn how to find the coefficient ( b ) in the quadratic equation ( 2x^2 + bx + c = 0 ) when roots are 3 and -5 using sum of roots and coefficient relationships."]
