1. The problem is to solve the cubic equation $$x^3 - 9x^2 + 22x + 12 = 0$$. 2. To solve a cubic equation, one common method is to try to find rational roots using the Rational Root Theorem, which suggests possible roots are factors of the constant term divided by factors of the leading coefficient. Here, possible roots are factors of 12: $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$$. 3. Test these possible roots by substituting into the polynomial: - For $$x=1$$: $$1 - 9 + 22 + 12 = 26 \neq 0$$ - For $$x=2$$: $$8 - 36 + 44 + 12 = 28 \neq 0$$ - For $$x=3$$: $$27 - 81 + 66 + 12 = 24 \neq 0$$ - For $$x= -1$$: $$-1 - 9 - 22 + 12 = -20 \neq 0$$ - For $$x= -2$$: $$-8 - 36 - 44 + 12 = -76 \neq 0$$ - For $$x= -3$$: $$-27 - 81 - 66 + 12 = -162 \neq 0$$ - For $$x=4$$: $$64 - 144 + 88 + 12 = 20 \neq 0$$ - For $$x= -4$$: $$-64 - 144 - 88 + 12 = -284 \neq 0$$ - For $$x=6$$: $$216 - 324 + 132 + 12 = 36 \neq 0$$ - For $$x= -6$$: $$-216 - 324 - 132 + 12 = -660 \neq 0$$ - For $$x=12$$: $$1728 - 1296 + 264 + 12 = 708 \neq 0$$ - For $$x= -12$$: $$-1728 - 1296 - 264 + 12 = -3276 \neq 0$$ 4. Since none of these are roots, try polynomial division or synthetic division to factor the cubic. Let's try synthetic division with $$x=1$$ again carefully: Coefficients: 1 (for $$x^3$$), -9 (for $$x^2$$), 22 (for $$x$$), 12 (constant) Synthetic division by 1: - Bring down 1 - Multiply 1 * 1 = 1, add to -9 = -8 - Multiply 1 * -8 = -8, add to 22 = 14 - Multiply 1 * 14 = 14, add to 12 = 26 (remainder) Remainder is 26, so 1 is not a root. Try synthetic division with $$x=2$$: - Bring down 1 - Multiply 2 * 1 = 2, add to -9 = -7 - Multiply 2 * -7 = -14, add to 22 = 8 - Multiply 2 * 8 = 16, add to 12 = 28 (remainder) Remainder is 28, so 2 is not a root. Try synthetic division with $$x= -1$$: - Bring down 1 - Multiply -1 * 1 = -1, add to -9 = -10 - Multiply -1 * -10 = 10, add to 22 = 32 - Multiply -1 * 32 = -32, add to 12 = -20 (remainder) Remainder is -20, so -1 is not a root. Try synthetic division with $$x= -2$$: - Bring down 1 - Multiply -2 * 1 = -2, add to -9 = -11 - Multiply -2 * -11 = 22, add to 22 = 44 - Multiply -2 * 44 = -88, add to 12 = -76 (remainder) Remainder is -76, so -2 is not a root. Try synthetic division with $$x=3$$: - Bring down 1 - Multiply 3 * 1 = 3, add to -9 = -6 - Multiply 3 * -6 = -18, add to 22 = 4 - Multiply 3 * 4 = 12, add to 12 = 24 (remainder) Remainder is 24, so 3 is not a root. Try synthetic division with $$x= -3$$: - Bring down 1 - Multiply -3 * 1 = -3, add to -9 = -12 - Multiply -3 * -12 = 36, add to 22 = 58 - Multiply -3 * 58 = -174, add to 12 = -162 (remainder) Remainder is -162, so -3 is not a root. Try synthetic division with $$x=4$$: - Bring down 1 - Multiply 4 * 1 = 4, add to -9 = -5 - Multiply 4 * -5 = -20, add to 22 = 2 - Multiply 4 * 2 = 8, add to 12 = 20 (remainder) Remainder is 20, so 4 is not a root. Try synthetic division with $$x= -4$$: - Bring down 1 - Multiply -4 * 1 = -4, add to -9 = -13 - Multiply -4 * -13 = 52, add to 22 = 74 - Multiply -4 * 74 = -296, add to 12 = -284 (remainder) Remainder is -284, so -4 is not a root. Try synthetic division with $$x=6$$: - Bring down 1 - Multiply 6 * 1 = 6, add to -9 = -3 - Multiply 6 * -3 = -18, add to 22 = 4 - Multiply 6 * 4 = 24, add to 12 = 36 (remainder) Remainder is 36, so 6 is not a root. Try synthetic division with $$x= -6$$: - Bring down 1 - Multiply -6 * 1 = -6, add to -9 = -15 - Multiply -6 * -15 = 90, add to 22 = 112 - Multiply -6 * 112 = -672, add to 12 = -660 (remainder) Remainder is -660, so -6 is not a root. Try synthetic division with $$x=12$$: - Bring down 1 - Multiply 12 * 1 = 12, add to -9 = 3 - Multiply 12 * 3 = 36, add to 22 = 58 - Multiply 12 * 58 = 696, add to 12 = 708 (remainder) Remainder is 708, so 12 is not a root. Try synthetic division with $$x= -12$$: - Bring down 1 - Multiply -12 * 1 = -12, add to -9 = -21 - Multiply -12 * -21 = 252, add to 22 = 274 - Multiply -12 * 274 = -3288, add to 12 = -3276 (remainder) Remainder is -3276, so -12 is not a root. 5. Since no rational roots are found, use the cubic formula or numerical methods. The cubic formula is complex, but we can find approximate roots using the depressed cubic method or numerical approximation. 6. Alternatively, use the substitution $$x = y + \frac{9}{3} = y + 3$$ to remove the quadratic term: Substitute $$x = y + 3$$: $$ (y+3)^3 - 9(y+3)^2 + 22(y+3) + 12 = 0 $$ Expand: $$ y^3 + 3 \cdot y^2 \cdot 3 + 3 \cdot y \cdot 3^2 + 3^3 - 9(y^2 + 2 \cdot 3 y + 3^2) + 22 y + 66 + 12 = 0 $$ Simplify: $$ y^3 + 9 y^2 + 27 y + 27 - 9 y^2 - 54 y - 81 + 22 y + 78 = 0 $$ Combine like terms: $$ y^3 + (9 y^2 - 9 y^2) + (27 y - 54 y + 22 y) + (27 - 81 + 78) = 0 $$ $$ y^3 - 5 y + 24 = 0 $$ 7. Now solve the depressed cubic $$y^3 - 5 y + 24 = 0$$. 8. Use the discriminant $$\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3$$ where $$p = -5$$ and $$q = 24$$: $$ \Delta = \left(\frac{24}{2}\right)^2 + \left(\frac{-5}{3}\right)^3 = 12^2 + \left(-\frac{5}{3}\right)^3 = 144 + \left(-\frac{125}{27}\right) = 144 - 4.6296 = 139.3704 > 0 $$ 9. Since $$\Delta > 0$$, there is one real root and two complex roots. 10. The real root is given by: $$ y = \sqrt[3]{-\frac{q}{2} + \sqrt{\Delta}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\Delta}} $$ Calculate: $$ -\frac{q}{2} = -12 $$ $$ \sqrt{\Delta} = \sqrt{139.3704} \approx 11.8 $$ So: $$ y = \sqrt[3]{-12 + 11.8} + \sqrt[3]{-12 - 11.8} = \sqrt[3]{-0.2} + \sqrt[3]{-23.8} $$ Calculate cube roots: $$ \sqrt[3]{-0.2} \approx -0.5848 $$ $$ \sqrt[3]{-23.8} \approx -2.8845 $$ Sum: $$ y \approx -0.5848 - 2.8845 = -3.4693 $$ 11. Recall $$x = y + 3$$, so: $$ x \approx -3.4693 + 3 = -0.4693 $$ 12. The other two roots are complex and can be found using the cubic formula or numerical methods, but the problem likely expects the real root. **Final answer:** The real root of the equation $$x^3 - 9x^2 + 22x + 12 = 0$$ is approximately $$x \approx -0.4693$$.