A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and -2. If \( a = 1 \), find the values of \( b \) and \( c \).
["Title: Solving a Quadratic Equation with Known Roots: Find b and c When a = 1 and Roots Are 3 and –2", "Understanding quadratic equations is fundamental in algebra, and knowing how roots determine coefficients like ( b ) and ( c ) simplifies solving real-world problems. In this article, we explore how to find the values of ( b ) and ( c ) for a quadratic equation of the form ( ax^2 + bx + c = 0 ), given its roots and coefficient ( a = 1 ) — specifically when the roots are ( 3 ) and ( -2 ).", "---", "### What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation written in the standard form:", "[\nax^2 + bx + c = 0\n]", "This equation has two roots (solutions), which can be real or complex. If the roots are known, say ( x_1 ) and ( x_2 ), and ( a ) is the leading coefficient, the equation can be written using roots as:", "[\na(x - x_1)(x - x_2) = 0\n]", "Expanding this formulation gives the standard form with coefficients ( a ), ( b ), and ( c ) in terms of the roots.", "---", "### Given Information", "We are told:", "- The roots are ( x_1 = 3 ) and ( x_2 = -2 )\n- The leading coefficient ( a = 1 )", "We are to determine the values of ( b ) and ( c ).", "---", "### Step 1: Use the Root-Factored Form", "From the roots, the equation becomes:", "[\na(x - 3)(x + 2) = 0\n]", "Since ( a = 1 ), this simplifies to:", "[\n(x - 3)(x + 2) = 0\n]", "---", "### Step 2: Expand the Factored Form", "Now expand the product:", "[\n(x - 3)(x + 2) = x(x + 2) - 3(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6\n]", "So the expanded quadratic equation is:", "[\nx^2 - x - 6 = 0\n]", "---", "### Step 3: Match to Standard Form", "Compare with ( ax^2 + bx + c = 0 ):", "[\nx^2 - x - 6 = 0\n\Rightarrow a = 1, \quad b = -1, \quad c = -6\n]", "---", "### Final Answer", "Thus, the values are:", "[\nb = -1, \quad c = -6\n]", "---", "### Why This Matters", "This method leverages the powerful relationship between roots and coefficients, enabling quick reconstruction of quadratics from roots. Whether you're solving physics problems, modeling financial growth, or analyzing engineering systems, knowing how to extract ( b ) and ( c ) from roots saves time and prevents errors.", "Remember: For a quadratic with roots ( x_1 ) and ( x_2 ) and leading coefficient ( a ):", "[\nb = -a(x_1 + x_2), \quad c = a(x_1 \cdot x_2)\n]", "Plugging in ( a = 1 ), ( x_1 = 3 ), ( x_2 = -2 ):", "[\nb = -(3 + (-2)) = -1, \quad c = 3 \ imes (-2) = -6\n]", "Same result, same logic — efficient and reliable.", "---", "Keywords: quadratic equation, roots 3 and -2, find b and c, given a=1, solve ax²+bx+c=0, algebra tips, math tutorial, quadratic roots formula, mathematics education."]
