Posons équation : 2x³ + 8x² = 384

["# Solving the Equation: 2x³ + 8x² = 384 – A Step-by-Step Guide", "Understanding how to solve cubic equations is essential in algebra, and one commonly encountered problem is 2x³ + 8x² = 384. This article guides you through solving this equation with clear, easy-to-follow steps—perfect for students, educators, or math enthusiasts aiming to master cubic equation solving techniques.", "---", "## What is the Equation?", "We start with the equation:", "[\n2x^3 + 8x^2 = 384\n]", "This is a cubic equation in standard form. Our goal is to find the real or complex solutions for ( x ) that satisfy this equation.", "---", "## Step 1: Rewrite the Equation in Standard Form", "First, bring all terms to one side to form an equation equal to zero:", "[\n2x^3 + 8x^2 - 384 = 0\n]", "---", "## Step 2: Simplify the Equation", "Divide every term by 2 to simplify solving:", "[\nx^3 + 4x^2 - 192 = 0\n]", "Now we seek to solve:", "[\nx^3 + 4x^2 - 192 = 0\n]", "---", "## Step 3: Attempt Factoring by Rational Root Theorem", "Using the Rational Root Theorem, possible rational roots are factors of 192 divided by factors of 1 (leading coefficient), so possible integer roots include ( \pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \ldots )", "Test ( x = 4 ):", "[\n(4)^3 + 4(4)^2 - 192 = 64 + 64 - 192 = -64 \quad \ ext{(Too low)}\n]", "Test ( x = 6 ):", "[\n6^3 + 4(6)^2 = 216 + 144 = 360 <br/>\neq 384 \quad \ ext{Too low}\n]", "Test ( x = 6 ) in original equation:\n( 2(6)^3 + 8(6)^2 = 2(216) + 8(36) = 432 + 288 = 720 ) — Way too high.", "Wait—let’s check ( x = 4 ) again in original form:", "Original equation:\n[\n2x^3 + 8x^2 = 384\n]", "At ( x = 4 ):\n[\n2(64) + 8(16) = 128 + 128 = 256 \quad \ ext{Still less than 384}\n]", "At ( x = 6 ):\n[\n2(216) + 8(36) = 432 + 288 = 720 \quad \ ext{Too big}\n]", "Try ( x = 5 ):", "[\n2(125) + 8(25) = 250 + 200 = 450 \quad \ ext{Still too high}\n]", "Try ( x = 4.5 ):", "[\n2(91.125) + 8(20.25) = 182.25 + 162 = 344.25\n]", "Still too low.", "Try ( x = 4.8 ):", "[\nx^2 = 23.04, \quad x^3 = 110.592\n]", "[\n2x^3 = 221.184, \quad 8x^2 = 8 \ imes 23.04 = 184.32\n]", "Sum = 221.184 + 184.32 = 405.504 — too high.", "So solution lies between 4.5 and 4.8.", "But let’s find exact solutions, not just numerical approximation.", "---", "## Step 4: Use Substitution to Reduce Degree", "Let’s use substitution to reduce the cubic.", "Start with:\n[\nx^3 + 4x^2 = 192\n]", "Complete the square for the quadratic term:", "Factor out coefficient of ( x^2 ):", "[\nx^2(x + 4) = 192\n]", "Not directly factorable, so try substitution:", "Let ( y = x + \frac{4}{3} ), to eliminate the ( x^2 ) term via substitution (standard depressed cubic method). However, a simpler method suitable for this equation is factoring by grouping after simplifying.", "But note: this cubic is not easily factorable by elementary means. Instead, we can use the rational root approach more carefully, or recognize that integer roots may be rare.", "Alternatively, use the cubic formula, or numerical methods, but since we seek crisp algebraic understanding, let’s explore factorization.", "---", "## Step 5: Try Factorization via Grouping (Attempt)", "Rewriting:", "[\nx^3 + 4x^2 - 192 = 0\n]", "Try factoring by grouping:\nWe look to write as ( (x^3 + ax^2) + (bx^2 - 192) )", "Alternatively, group as ( x^2(x + 4) = 192 ) — still not helpful algebraically.", "---", "## Step 6: Use Numerical Methods or Graphing Insight (Optional for Accuracy)", "Since analytical factoring is challenging, we observe from earlier trials:", "- At ( x = 5 ): LHS = 450\n- At ( x = 4.5 ): LHS = 344.25\n- At ( x = 4.7 ):", "( x^2 = 22.09, x^3 = 103.823 )\n( 2x^3 = 207.646, 8x^2 = 176.72 ) → Sum = 384.366 — very close!", "Try ( x = 4.69 ):", "( x^2 = 21.9961 ), ( x^3 = 4.69 \ imes 21.9961 \approx 103.25 )\n( 2x^3 \approx 206.5 ), ( 8x^2 \approx 175.97 ) → ≈ 382.47", "So actual root ≈ 4.702", "But since this is a school-level equation, expect one real root and two complex conjugates.", "---", "## Step 7: Use Cubic Formula (Brief Overview) — Advanced Technique", "For the depressed cubic form, we can apply Cardano’s method, but that is lengthy.", "Instead, note:", "Divide original simplified cubic:", "[\nx^3 + 4x^2 - 192 = 0\n]", "Use substitution ( x = y - \frac{4}{3} ) to eliminate the ( x^2 ) term.", "Let:", "[\nx = y - \frac{4}{3}\n]", "Then:", "[\nx^3 = \left(y - \frac{4}{3}\right)^3 = y^3 - 4y^2 + \frac{16}{3}y - \frac{64}{27}\n]\n[\n4x^2 = 4\left(y - \frac{4}{3}\right)^2 = 4\left(y^2 - \frac{8}{3}y + \frac{16}{9}\right) = 4y^2 - \frac{32}{3}y + \frac{64}{9}\n]", "Add:", "[\nx^3 + 4x^2 = y^3 - 4y^2 + \frac{16}{3}y - \frac{64}{27} + 4y^2 - \frac{32}{3}y + \frac{64}{9}\n]", "Simplify:", "- ( y^2 ) terms cancel: ( -4y^2 + 4y^2 = 0 )\n- ( y ) terms: ( \frac{16}{3} - \frac{32}{3} = -\frac{16}{3}y )\n- Constants: ( -\frac{64}{27} + \frac{64}{9} = -\frac{64}{27} + \frac{192}{27} = \frac{128}{27} )", "So:", "[\nx^3 + 4x^2 = y^3 - \frac{16}{3}y + \frac{128}{27}\n]", "Set equal to 192:", "[\ny^3 - \frac{16}{3}y + \frac{128}{27} = 192\n]", "[\ny^3 - \frac{16}{3}y = 192 - \frac{128}{27} = \frac{5184 - 128}{27} = \frac{5056}{27}\n]", "Multiply through by 27:", "[\n27y^3 - 144y = 5056\n]", "[\n27y^3 - 144y - 5056 = 0\n]", "Now solve ( 27y^3 - 144y - 5056 = 0 ) — still complex.", "This confirms no simple rational root exists.", "---", "## Step 8: Conclusion — Exact and Approximate Solutions", "No rational root exists. The real solution lies near ( x \approx 4.702 ). Approximate solutions:", "- One real root: ( x \approx 4.702 )\n- Two complex conjugate roots (from cubic nature)", "Using numerical solvers or graphing calculators, the precise real root is approximately:", "[\nx \approx 4.702\n]", "For exact form, we use the cubic formula:", "Let ( a = 1, b = 4, c = 0, d = -192 )", "Cardano’s method yields depressed cubic and resolves, but output is messy.", "---", "## Practical Takeaway", "The equation 2x³ + 8x² = 384 simplifies to the cubic:", "[\nx^3 + 4x^2 - 192 = 0\n]", "Which does not factor nicely with integers. Solving it exactly requires advanced algebra, but numerically:", "[\n\boxed{x \approx 4.702}\n]", "is the real solution.", "---", "## Tips for Solving Similar Cubic Equations:", "1. Simplify first: Divide by common factor (here, 2).\n2. Check for rational roots: Try small integers.\n3. Use substitution: ( x = y - \frac{b}{3a} ) to eliminate ( x^2 ).\n4. Estimate solutions: Use graphing or numerical methods when factoring fails.\n5. Verify: Plug values back into original equation.", "---", "## Further Reading:", "- Solving Cubic Equations Step-by-Step\n- Depressed Cubic Forms and Cardano’s Formula\n- Applications of Cubic Equations in Physics and Engineering", "---", "## Keywords for SEO:", "- Solve 2x³ + 8x² = 384\n- Cubic equation solution\n- How to solve x³ + 4x² = 192\n- Algebraic equation 2x³ + 8x² = 384\n- Real root of cubic equation\n- Solving cubic equations by substitution\n- Mathematics tutorial cubic equations", "---", "Transform abstract algebra into practical mastery — now you understand not just how to solve, but why and when such techniques matter."]