["# Solving the Quadratic Equation: ( x(20 - x) = 96 )", "If you're looking to solve straightforward quadratic equations, equation ( x(20 - x) = 96 ) offers an accessible and instructive example. This article explores how to transform and solve this equation, explaining each step clearly. Whether you're a high school student mastering algebra or someone brushing up on key math concepts, understanding how to solve ( x(20 - x) = 96 ) helps build strong foundational skills.", "---", "## Step-by-Step Solution to ( x(20 - x) = 96 )", "### 1. Expand the Equation", "Start by distributing ( x ) across the parentheses:", "[
\nx(20 - x) = 96
\n]
\n[
\n20x - x^2 = 96
\n]", "### 2. Rearrange into Standard Quadratic Form", "Move all terms to one side to form a standard quadratic equation ( ax^2 + bx + c = 0 ):", "[
\n20x - x^2 - 96 = 0
\n]
\n[
\n-x^2 + 20x - 96 = 0
\n]", "Multiply the entire equation by (-1) to simplify the leading coefficient:", "[
\nx^2 - 20x + 96 = 0
\n]", "Now the equation is in standard form:
\n( x^2 - 20x + 96 = 0 )", "### 3. Factor or Use the Quadratic Formula", "This equation is factorable—look for two numbers that multiply to 96 and add to -20. These are (-12) and (-8):", "[
\n(x - 12)(x - 8) = 0
\n]", "Setting each factor equal to zero gives:", "[
\nx - 12 = 0 \quad \Rightarrow \quad x = 12
\n]
\n[
\nx - 8 = 0 \quad \Rightarrow \quad x = 8
\n]", "---", "## Interpretation of the Solutions", "The solutions ( x = 8 ) and ( x = 12 ) represent the values that satisfy the original equation. Graphically, these points correspond to the ( x )-coordinates where the parabola ( y = -x^2 + 20x - 96 ) intersects the ( x )-axis.", "From a real-world perspective, this type of equation often arises in optimization problems or area calculations, especially when modeling a rectangular space with a fixed perimeter.", "---", "## Why This Equation Matters", "The equation ( x(20 - x) = 96 ) illustrates key algebraic and analytical skills:", "- Expanding expressions: Turning product forms into standard quadratic form.
\n- Simplifying equations: Using coefficient rules to standardize quadratic expressions.
\n- Solving quadratics: Applying factoring and the zero-product property to find roots.
\n- Interpreting results: Understanding what real values satisfy the equation.", "These are cornerstones in algebra, essential for higher-level math and STEM applications.", "---", "## Bonus: Solving Using the Quadratic Formula", "For completeness, here’s how the quadratic formula applies:", "From ( x^2 - 20x + 96 = 0 ),
\n( a = 1 ), ( b = -20 ), ( c = 96 ).", "Using ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), compute the discriminant:", "[
\nb^2 - 4ac = (-20)^2 - 4(1)(96) = 400 - 384 = 16
\n]", "Then:", "[
\nx = \frac{20 \pm \sqrt{16}}{2} = \frac{20 \pm 4}{2}
\n]
\n[
\nx = \frac{24}{2} = 12 \quad \ ext{or} \quad x = \frac{16}{2} = 8
\n]", "Same results—confirming accuracy.", "---", "## Summary", "The equation ( x(20 - x) = 96 ) represents a simple yet powerful quadratic problem. By expanding, rearranging, factoring, and solving, we find the solutions ( x = 8 ) and ( x = 12 ). Mastering such problems strengthens your algebra foundation, prepares you for advanced equations, and enhances problem-solving abilities critical across many fields.", "If you’re ready to practice, try similar problems like ( x(30 - x) = 256 ) or explore applications in geometry and finance—where this equation type frequently appears.", "---", "Keywords: solve ( x(20 - x) = 96 ), quadratic equation, algebra, solving quadratics, factoring, real-world applications, quadratic form, step-by-step solution, electronic learning, math tutorial.", "---", "Start solving quadratic equations today—understanding ( x(20 - x) = 96 ) is a key stepping stone!"]
