A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and 5. If \( a = 2 \), find \( b \) and \( c \).
["Understanding Quadratic Equations: Finding ( b ) and ( c ) When Roots Are Known (With ( a = 2 ))", "A fundamental concept in algebra is the quadratic equation, typically expressed as:", "[\nax^2 + bx + c = 0\n]", "When the roots of this equation are known, a powerful relationship exists between the coefficients and the roots—rooted in Vieta’s formulas. This article explores how to determine the coefficients ( b ) and ( c ) when a quadratic has specific roots, using a practical example with ( a = 2 ), and roots 3 and 5.", "---", "### The Role of Roots in Quadratic Equations", "If ( r_1 ) and ( r_2 ) are the roots of the quadratic equation ( ax^2 + bx + c = 0 ), then the equation can be rewritten using its factored form:", "[\na(x - r_1)(x - r_2) = 0\n]", "Expanding this gives:", "[\na(x - r_1)(x - r_2) = a[x^2 - (r_1 + r_2)x + r_1 r_2]\n]", "From here, we identify:\n- Sum of roots: ( r_1 + r_2 = -\frac{b}{a} )\n- Product of roots: ( r_1 \cdot r_2 = \frac{c}{a} )", "Using these formulas, we can solve for ( b ) and ( c ) even when ( a = 2 ).", "---", "### Given Values", "- Roots: ( r_1 = 3 ), ( r_2 = 5 )\n- Leading coefficient: ( a = 2 )\n- Goal: Find ( b ) and ( c )", "---", "### Step 1: Compute the Sum and Product of Roots", "Calculate the sum and product:\n[\nr_1 + r_2 = 3 + 5 = 8\n]\n[\nr_1 \cdot r_2 = 3 \cdot 5 = 15\n]", "---", "### Step 2: Use Vieta’s Formulas to Find ( b ) and ( c )", "From Vieta’s relations with ( a = 2 ):\n[\nr_1 + r_2 = -\frac{b}{a} \implies 8 = -\frac{b}{2} \implies b = -16\n]\n[\nr_1 r_2 = \frac{c}{a} \implies 15 = \frac{c}{2} \implies c = 30\n]", "---", "### Step 3: Write the Final Quadratic Equation", "Substituting the values:\n[\n2x^2 - 16x + 30 = 0\n]", "---", "### Why This Matters in Real-World Applications", "Finding ( b ) and ( c ) from roots is not just an academic exercise. Engineers, data scientists, and physicists often rely on quadratic models to predict behavior, optimize systems, and analyze data. When the roots represent critical points (like time when a projectile hits the ground), knowing the coefficients provides full control over the model.", "---", "### Summary", "If a quadratic equation ( 2x^2 + bx + c = 0 ) has roots 3 and 5:\n- The sum of roots is ( 8 ), so ( b = -2 \ imes 8 = -16 )\n- The product of roots is ( 15 ), so ( c = 2 \ imes 15 = 30 )", "Thus, the quadratic equation becomes:", "[\n\boxed{2x^2 - 16x + 30 = 0}\n]", "Understanding how roots determine coefficients helps unlock deeper algebraic insights and practical problem-solving skills. Whether you're solving equations or modeling real life, mastering this concept is essential.", "---", "Key SEO Keywords: \nquadratic equation roots, find b and c quadratic, quadratic roots formula, find b and c from roots, ( ax^2 + bx + c = 0 ) roots 3 and 5, quadratic coefficients from roots, Vieta’s formulas for quadratics, solve quadratic using root sum and product, ( a = 2 ) quadratic equation, step-by-step quadratic roots", "Meta Title:\nHow to find ( b ) and ( c ) in ( 2x^2 + bx + c = 0 ) when roots are 3 and 5 — Step-by-step solution", "Meta Description:\nLearn how to determine coefficients ( b ) and ( c ) in the quadratic equation ( 2x^2 + bx + c = 0 ) when roots are 3 and 5. Use Vieta’s formulas to compute ( b = -16 ) and ( c = 30 )."]
