A quadratic equation \( ax^2 + bx + c = 0 \) has roots 4 and -3. If \( a = 1 \), what are the values of \( b \) and \( c \)?
["Understanding Quadratic Equations: Finding ( b ) and ( c ) from Known Roots", "A quadratic equation takes the general form ( ax^2 + bx + c = 0 ), and one of its most powerful features is that its roots reveal key information about the equation’s coefficients. In this article, we explore how to determine the values of ( b ) and ( c ) when two roots are known, specifically when the roots are ( x = 4 ) and ( x = -3 ), and the leading coefficient ( a = 1 ).", "---", "### Why Roots Matter in Quadratic Equations", "When a quadratic equation has real roots ( r_1 ) and ( r_2 ), it can be rewritten in factored form using the fact that:", "[\nax^2 + bx + c = a(x - r_1)(x - r_2)\n]", "For a monic equation (where ( a = 1 )), this simplifies to:", "[\nx^2 + bx + c = (x - 4)(x + 3)\n]", "Expanding this product gives a quadratic equation in standard form, from which we can clearly identify ( b ) and ( c ).", "---", "### Step-by-Step: Calculating ( b ) and ( c )", "Given the roots are ( x = 4 ) and ( x = -3 ), and ( a = 1 ), proceed as follows:", "1. Write the factored form:", "[\nx^2 + bx + c = (x - 4)(x + 3)\n]", "2. Expand the product using the distributive property (FOIL method):", "[\n(x - 4)(x + 3) = x \cdot x + x \cdot 3 - 4 \cdot x - 4 \cdot 3\n]\n[\n= x^2 + 3x - 4x - 12\n]\n[\n= x^2 - x - 12\n]", "3. Compare with the standard form ( x^2 + bx + c ):", "- Coefficient of ( x ): ( b = -1 )\n- Constant term: ( c = -12 )", "---", "### Final Answer", "If ( ax^2 + bx + c = 0 ) has roots ( 4 ) and ( -3 ) and ( a = 1 ), then:", "- ( b = -1 )\n- ( c = -12 )", "So, the quadratic equation is:", "[\nx^2 - x - 12 = 0\n]", "---", "### Why This Method Works", "This approach leverages the deep connection between the roots of a quadratic and its coefficients — a relationship foundational in algebra. Whether solving equations, modeling real-world phenomena, or teaching quadratic functions, mastering this technique helps students rapidly convert roots into explicit coefficients, making equations easier to analyze and solve.", "---", "Keywords: quadratic equation, roots 4 and -3, quadratic formula, coefficients from roots, ( ax^2 + bx + c ), ( a = 1 ), find ( b ) and ( c ), algebraic derivation, polynomial factoring."]
