r^3 - (r^3 - 6r^2 + 12r - 8) = 96

["# Mastering the Equation: Solving ( r^3 - (r^3 - 6r^2 + 12r - 8) = 96 )", "Equations involving cubic expressions can seem intimidating at first glance, but with the right approach, they become manageable—and even satisfying to solve. In this article, we’ll walk through the step-by-step solution to the equation:", "[\nr^3 - (r^3 - 6r^2 + 12r - 8) = 96\n]", "---", "## Simplifying the Left-Hand Side", "Start by simplifying the expression on the left:", "[\nr^3 - (r^3 - 6r^2 + 12r - 8) = r^3 - r^3 + 6r^2 - 12r + 8\n]", "Simplify term by term:", "[\n= 0 + 6r^2 - 12r + 8 = 6r^2 - 12r + 8\n]", "Now the equation becomes:", "[\n6r^2 - 12r + 8 = 96\n]", "---", "## Bringing All Terms to One Side", "Subtract 96 from both sides to set the equation to zero:", "[\n6r^2 - 12r + 8 - 96 = 0\n]", "[\n6r^2 - 12r - 88 = 0\n]", "---", "## Simplifying the Quadratic Equation", "Divide the entire equation by 2 to simplify:", "[\n3r^2 - 6r - 44 = 0\n]", "Now we solve this quadratic using the quadratic formula:", "[\nr = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 3 ), ( b = -6 ), and ( c = -44 ). Plug in the values:", "[\nr = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(-44)}}{2(3)}\n]", "[\nr = \frac{6 \pm \sqrt{36 + 528}}{6} = \frac{6 \pm \sqrt{564}}{6}\n]", "---", "## Simplify the Square Root Term", "Find the square root:", "[\n\sqrt{564} = \sqrt{4 \cdot 141} = 2\sqrt{141}\n]", "So the solution becomes:", "[\nr = \frac{6 \pm 2\sqrt{141}}{6} = \frac{2(3 \pm \sqrt{141})}{6} = \frac{3 \pm \sqrt{141}}{3}\n]", "---", "## Final Solutions", "Thus, the exact solutions are:", "[\nr = \frac{3 + \sqrt{141}}{3} \quad \ ext{and} \quad r = \frac{3 - \sqrt{141}}{3}\n]", "These are the two real roots of the original cubic equation after simplification.", "---", "## Why This Equation Matters", "While this equation may appear purely academic, solving nested cubic expressions like this strengthens algebraic intuition and prepares you for more complex real-world modeling in physics, engineering, and computer science. Simplifying expressions before solving is a critical skill, as seen here when canceling ( r^3 ) terms led directly to a manageable quadratic.", "---", "## How to Find Exact Roots More Efficiently", "Instead of manually simplifying, tools like symbolic math software (e.g., WolframAlpha, SymPy) can verify your steps or solve it instantly. However, understanding each transformation ensures you master the underlying math—not just the formula.", "---", "## Conclusion", "The equation\n[\nr^3 - (r^3 - 6r^2 + 12r - 8) = 96\n]\nreduces neatly to a quadratic, revealing two elegant real solutions involving the square root of 141. By methodically simplifying and applying the quadratic formula, we’ve uncovered the root(s) with confidence. Whether for homework, coding, or problem-solving practice, mastering this process builds analytical power one equation at a time.", "---", "### Keywords:\ncubic equation, solve cubic equation, simplify algebraic expressions, quadratic formula, math problem solving, ( r^3 - 6r^2 + 12r - 8 = 96 ), algebra tutorial, STEM problem solving", "---", "Ready to tackle your next equation with clarity? Start by simplifying, then breaking down—your next breakthrough awaits."]