\( 20x - x^2 = 96 \) → \( x^2 - 20x + 96 = 0 \).
["# Solving the Quadratic Equation: ( 20x - x^2 = 96 )", "When faced with the equation ( 20x - x^2 = 96 ), it’s important to rearrange it into a standard quadratic form to apply solution methods effectively. Rearranging the equation leads to:", "[\nx^2 - 20x + 96 = 0\n]", "## Why Rewrite the Equation?", "Switching to the standard form ( ax^2 + bx + c = 0 ) allows you to use well-known techniques such as factoring, completing the square, or applying the quadratic formula. This renormalization makes the equation easier to analyze and solve.", "## Step-by-Step Solution", "### Step 1: Rearranging Terms", "Start with:\n[\n20x - x^2 = 96\n]", "Subtract 96 from both sides:\n[\n-x^2 + 20x - 96 = 0\n]", "To avoid negative leading coefficient, multiply the entire equation by –1:\n[\nx^2 - 20x + 96 = 0\n]", "### Step 2: Factor the Quadratic", "We now solve:\n[\nx^2 - 20x + 96 = 0\n]", "We seek two numbers that multiply to ( 96 ) and add to ( -20 ). These numbers are ( -12 ) and ( -8 ), because:", "[\n-12 \ imes -8 = 96 \quad \ ext{and} \quad -12 + (-8) = -20\n]", "Thus, the factored form is:\n[\n(x - 12)(x - 8) = 0\n]", "### Step 3: Solve Using the Zero Product Property", "Set each factor equal to zero:\n[\nx - 12 = 0 \quad \Rightarrow \quad x = 12\n]\n[\nx - 8 = 0 \quad \Rightarrow \quad x = 8\n]", "## Final Answer", "The solutions to the equation ( 20x - x^2 = 96 ) are:\n[\nx = 8 \quad \ ext{and} \quad x = 12\n]", "## Key Takeaways", "- Rearranging equations to standard quadratic form is crucial for applying root-finding methods.\n- Factoring is efficient here due to integer solutions, but methods like the quadratic formula (if factoring is difficult) remain reliable.\n- Verify solutions by substituting back into the original equation:\n - For ( x = 8 ): ( 20(8) - 8^2 = 160 - 64 = 96 ) ✅\n - For ( x = 12 ): ( 20(12) - 12^2 = 240 - 144 = 96 ) ✅", "## Related Topics", "- Quadratic equations and factoring techniques\n- Solving real-world problems using quadratics\n- Graphing quadratic functions – roots correspond to x-intercepts", "Use this guide to confidently solve any quadratic equation and deepen your algebra skills!", "---", "Keyword-rich SEO focus: "solve ( 20x - x^2 = 96 )", "quadratic equation ( x^2 - 20x + 96 = 0 )", "factor quadratic solutions", "quadratic formulas", "step-by-step quadratic solution"."]
