But check equation again: 2x³ + 8x² = 384 → x³ + 4x² - 192 = 0

["Solving the Cubic Equation: A Step-by-Step Guide to 2x³ + 8x² = 384 – (x³ + 4x² – 192) = 0", "Finding solutions to cubic equations like 2x³ + 8x² = 384 – (x³ + 4x² – 192) = 0 often confuses many students and math enthusiasts. But with the right approach, simplifying the equation and applying proper algebraic techniques, solving these cubic expressions becomes manageable and even straightforward. In this SEO-optimized article, we’ll walk you through checking and solving this equation step-by-step, ensuring clarity and accuracy.", "---", "### Why Understanding Cubic Equations Matters: SEO Relevance", "Cubic equations—polynomials of degree three—play an essential role in math, engineering, and physics. Articles about solving cubic equations, polynomial simplification, and algebraic problem-solving consistently rank high in search results, especially among students, teachers, and self-learners. Including key terms like “how to solve cubic equations,” “step-by-step cubic solution,” and “simplifying polynomial equations” boosts SEO visibility.", "Our goal is to provide a complete, optimized explanation of solving:\n2x³ + 8x² = 384 – (x³ + 4x² – 192) = 0", "---", "### Step 1: Clear the Equation Carefully", "Start by simplifying the right-hand side:", "[\n384 – (x³ + 4x² – 192) = 0\n]", "Distribute the negative sign inside the parentheses:", "[\n384 – x³ – 4x² + 192 = 0\n]", "Combine like terms:", "[\n(384 + 192) – x³ – 4x² = 0\n\quad\Rightarrow\quad\n576 – x³ – 4x² = 0\n]", "Rewriting in standard form (descending powers of (x)):", "[\n- x³ - 4x² + 576 = 0\n]", "Multiply both sides by –1 to make the leading coefficient positive (important for consistent solving):", "[\nx³ + 4x² - 576 = 0\n]", "📌 Key SEO Tip: Always rewrite equations in standard form (+x³ term), increase keyword density with phrases like “standard form cubic equation” and “solving cubic equations step-by-step.”", "---", "### Step 2: Simplify and Verify the Equation", "We now work with:\n[\nx³ + 4x² - 576 = 0\n]", "This cubic equation can be solved using:", "- Rational Root Theorem (to test possible rational roots),\n- Factorization,\n- Or numerical methods for more complex cases.", "Let’s test possible rational roots from factors of 576.", "---", "### Step 3: Apply the Rational Root Theorem", "Possible rational roots are factors of 576 divided by factors of 1 (leading coefficient), so test divisors of 576.", "Try (x = 6):\n[\n6³ + 4(6²) - 576 = 216 + 144 - 576 = -216 ≠ 0\n]", "Try (x = 8):\n[\n8³ + 4(8²) - 576 = 512 + 256 - 576 = 192 ≠ 0\n]", "Try (x = -12):\n[\n(-12)^3 + 4(-12)^2 - 576 = -1728 + 576 - 576 = -1728 ≠ 0\n]", "Try (x = 6) again — already tried. Try (x = 6) might not work — instead try (x = 6) again symbolically.", "Wait — try (x = 6) carefully:\n(6³ = 216), (4×6² = 4×36 = 144),\n216 + 144 = 360, 360 – 576 = –216 ≠ 0.", "Try (x = 8):\n8³ = 512, 4×64 = 256 → 512 + 256 = 768 > 576. Too big.", "Try (x = 6) again? No. Try smaller positive integer.", "Try (x = 6): no. Try (x = 4):\n64 + 64 – 576 = –448\nTry (x = 8): too big\nTry (x = 9):\n729 + 4×81 = 729 + 324 = 1053 – 576 = 477 ≠ 0\nWait — try (x = 6) → 360 – 576 = –216\nTry (x = 8): 512 + 256 = 768 – 576 = 192\nTry (x = 7):\n(7³ = 343), (4×49 = 196), total = 343 + 196 = 539 – 576 = –37\nTry (x = 7.5):\n7.5³ = 421.875, 4×56.25 = 225 → 421.875 + 225 = 646.875 – 576 = 70.875", "Root is between 7 and 8 — possibly irrational.", "But let’s return and rearrange instead.", "---", "### Step 4: Use Substitution or Factor by Grouping (if possible)", "Notice the simplified equation is:", "[\nx³ + 4x² - 576 = 0\n]", "Try rational root candidates: ±1, 2, 3, 4, 6, 8, 9, 12, 16, ... divided by 1.", "Try (x = 6): still –216\nTry (x = 8): 512 + 256 – 576 = 192\nTry (x = 6.5):\n6.5³ = 274.625, 4×(6.5)² = 4×42.25 = 169 → total = 274.625 + 169 = 443.625 – 576 = –132.375\nTry (x = 7.2):\n7.2³ ≈ 373.248, 4×(51.84) = 207.36 → 373.248 + 207.36 = 580.608 – 576 = +4.608\nClose! Try (x = 7.15):\n7.15³ ≈ 366.75, 4×51.1225 ≈ 204.49 → total ≈ 571.24 – 576 ≈ –4.76\nSo root ≈ 7.18 — irrational.", "But wait — perhaps factor?", "Try factoring by grouping — not directly possible.", "But let’s revisit simplification: was simplification correct?", "Original equation:\n[\n2x³ + 8x² = 384 - (x³ + 4x² - 192)\n]", "Right-hand side:\n384 – x³ – 4x² + 192 = (384 + 192) – x³ – 4x² = 576 – x³ – 4x²", "So:\n[\n2x³ + 8x² = 576 - x³ - 4x²\n]", "Move all terms to left:", "[\n2x³ + 8x² + x³ + 4x² - 576 = 0\n\Rightarrow 3x³ + 12x² - 576 = 0\n]", "🚨 Important correction here!\nEarlier step incorrectly simplified the right-hand side.", "Correct version:", "[\n2x³ + 8x² = 384 - (x³ + 4x² - 192)\n\Rightarrow\n2x³ + 8x² = 384 - x³ - 4x² + 192\n\Rightarrow\n2x³ + 8x² + x³ + 4x² - 576 = 0\n\Rightarrow\n3x³ + 12x² - 576 = 0\n]", "✅ So the correct equation is:\n[\n3x³ + 12x² - 576 = 0\n]", "---", "### Step 5: Divide bycommon factor to simplify", "Divide entire equation by 3:", "[\nx³ + 4x² - 192 = 0\n]", "This is the simplified correct form.", "---", "### Step 6: Solve the Simplified Cubic: (x³ + 4x² - 192 = 0)", "Now solve:", "[\nx³ + 4x² - 192 = 0\n]", "#### Try rational roots again: factors of 192 over 1.", "Try (x = 4):\n64 + 64 – 192 = 128 – 192 = –64\nTry (x = 6):\n216 + 144 – 192 = 360 – 192 = 168\nTry (x = 4) — no\nTry (x = -8):\n–512 + 256 – 192 = –448 ≠ 0\nTry (x = 8): 512 + 256 – 192 = 576 – 192 = 384 ≠ 0\nTry (x = 4) again — no\nTry (x = -6):\n–216 + 144 – 192 = –264 ≠ 0\nTry (x = 6): 360 – 192 = 168 ≠ 0", "Try (x = 4.5):\n91.125 + 4×20.25 = 91.125 + 81 = 172.125 – 192 ≈ –19.875\nTry (x ≈ 5): 125 + 100 – 192 = 225 – 192 = 33", "Root between 4.5 and 5 — irrational again.", "But now apply rational root attempt more precisely.", "Use numerical methods or Cardano’s formula, but for clarity, suppose we factor using one real root.", "Alternatively, use depressed cubic from here.", "---", "### Step 7: Use Substitution for Cubic Elimination", "Let (x = y - \dfrac{b}{3a}). For (x³ + 4x² + 0x - 192 = 0),\n(a = 1), (b = 4), so shift:", "Let (x = y - \frac{4}{3"]