w_1 + w_2 + w_3 = 8, \quad w_1w_2 + w_2w_3 + w_3w_1 = 9, \quad w_1w_2w_3 = 18.

["Understanding the Cubic Roots: Solving the Simultaneous Equations w₁ + w₂ + w₃ = 8, w₁w₂ + w₂w₃ + w₃w₁ = 9, w₁w₂w₃ = 18", "When faced with symmetric polynomial equations involving three unknowns, the elegant connection between roots and coefficients shines through — a classic application of Vieta’s formulas. Given the system:", "- ( w_1 + w_2 + w_3 = 8 )\n- ( w_1w_2 + w_2w_3 + w_3w_1 = 9 )\n- ( w_1w_2w_3 = 18 )", "These expressions correspond to the elementary symmetric sums that define the roots of a cubic polynomial. Specifically, if ( w_1, w_2, w_3 ) are roots, the cubic equation they satisfy is:", "[\nx^3 - (w_1 + w_2 + w_3)x^2 + (w_1w_2 + w_2w_3 + w_3w_1)x - w_1w_2w_3 = 0\n]", "Substituting the given values:", "[\nx^3 - 8x^2 + 9x - 18 = 0\n]", "Our task now reduces to solving this cubic equation to find ( w_1, w_2, w_3 ). By applying algebraic techniques such as the Rational Root Theorem, synthetic division, and factoring, we can identify the roots.", "---", "### Step 1: Rational Root Candidates", "The Rational Root Theorem suggests that any rational root of the polynomial must be a divisor of the constant term (18) divided by a divisor of the leading coefficient (1). So possible rational roots include:", "[\n\pm1, \pm2, \pm3, \pm6, \pm9, \pm18\n]", "We test these by substituting:", "- ( x = 1 ): ( 1 - 8 + 9 - 18 = -16 <br/>\ne 0 )\n- ( x = 2 ): ( 8 - 32 + 18 - 18 = -24 <br/>\ne 0 )\n- ( x = 3 ): ( 27 - 72 + 27 - 18 = -36 <br/>\ne 0 )\n- ( x = 6 ): ( 216 - 288 + 54 - 18 = -36 <br/>\ne 0 )\n- ( x = -1 ): ( -1 - 8 - 9 - 18 = -36 <br/>\ne 0 )\n- ( x = -2 ): ( -8 - 32 - 18 - 18 = -76 <br/>\ne 0 )", "None are roots. However, try ( x = 9 ):", "[\n9^3 - 8 \cdot 9^2 + 9 \cdot 9 - 18 = 729 - 648 + 81 - 18 = 144 <br/>\ne 0\n]", "Try ( x = 2 ) again with corrected calculation:\nWait — double-check ( x = 3 ):", "[\n3^3 = 27,\quad -8(9) = -72,\quad +9(3)=27,\quad -18 = -18\n27 - 72 = -45, \quad -45 + 27 = -18, \quad -18 -18 = -36 <br/>\ne 0\n]", "Try ( x = 6 ):", "[\n216 - 288 + 54 - 18 = (216 + 54) - (288 + 18) = 270 - 306 = -36\n]", "Try ( x = -1 ) again? Instead, consider that no rational root exists.", "---", "### Step 2: Using Initiative: Break the Cubic", "Since rational roots fail, apply the cubic formula approximation or numerical insight, but first verify whether all roots are real by analyzing the discriminant of the cubic.", "For a cubic ( x^3 + ax^2 + bx + c = 0 ), the discriminant ( \Delta ) determines root nature:", "[\n\Delta = 18abc - 4a^3c + a^2b^2 - 4b^3 - 27c^2\n]", "Our cubic: ( x^3 - 8x^2 + 9x - 18 ), so:", "- ( a = -8 ), ( b = 9 ), ( c = -18 )", "Compute:", "- ( 18abc = 18(-8)(9)(-18) = 18 \cdot 1296 = 23328 )\n- ( -4a^3c = -4(-512)(-18) = -4 \cdot 9216 = -36864 )\n- ( a^2b^2 = 64 \cdot 81 = 5184 )\n- ( -4b^3 = -4 \cdot 729 = -2916 )\n- ( -27c^2 = -27 \cdot 324 = -8748 )", "Sum:", "[\n\Delta = 23328 - 36864 + 5184 - 2916 - 8748 =\n(23328 + 5184) = 28512\n( -36864 -2916) = -39780, -8748 → total subtract: -48708\n28512 - 48708 = -20196 < 0\n]", "Since ( \Delta < 0 ), the cubic has one real root and two complex conjugate roots.", "---", "### Step 3: Approximate or Express the Real Root Exactly", "We seek one real root to factor the cubic:", "[\nx^3 - 8x^2 + 9x - 18 = 0\n]", "Use Cardano’s method or numerical estimation.", "Try ( x = 4 ):", "[\n64 - 128 + 36 - 18 = (64 + 36) = 100, (-128 - 18) = -146 → -46\n]", "Try ( x = 5 ):", "[\n125 - 200 + 45 - 18 = (125 + 45) = 170, (-200 -18) = -218 → -48\n]", "Try ( x = 6 ): already done → -36\nTry ( x = 3.5 ):\n( 3.5^3 = 42.875 ), ( 8 \cdot 12.25 = 98 ), ( 9 \cdot 3.5 = 31.5 )\n→ ( 42.875 - 98 + 31.5 - 18 = (42.875 + 31.5) = 74.375, (-98 -18) = -116 → -41.625 )", "Try ( x = 2.5 ):", "( 15.625 - 50 + 22.5 - 18 = (15.625 + 22.5) = 38.125, (-50 -18) = -68 → -29.875 )", "Try ( x = 1.5 ):", "( 3.375 - 18 + 13.5 - 18 = (3.375 + 13.5) = 16.875, (-18 -18) = -36 → -19.125 )", "Try ( x = 0.5 ):", "( 0.125 - 2 + 4.5 - 18 = (0.125 + 4.5) = 4.625, (-2 -18) = -20 → -15.375 )", "All negative — try higher than 6?", "Try ( x = 7 ):\n( 343 - 392 + 63 - 18 = (343 + 63) = 406, (-392 -18) = -410 → -4 )", "Close! Try ( x = 7.1 ):", "( 7.1^3 = 357.911 ), ( 8 \cdot 50.41 = 403.28 ), ( 9 \cdot 7.1 = 63.9 )\n→ ( 357.911 - 403.28 = -45.369 ), ( +63.9 = 18.531 ), ( -18 = 0.531 )", "So root between **"]