s^3 + 6s^2 + 15s + 12 = 0
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You asked:
Solve the equation
s 3 + 6 s 2 + 15 s + 12 = 0 {s}^{3} + 6 {s}^{2} + 15 s + 12 = 0 s 3 + 6 s 2 + 15 s + 12 = 0 for the variable
s s s .
MathBot Answer:
The 3 solutions to the equation are:
s = − 2 + 1 + 2 3 − 1 1 + 2 3 ≈ − 1.4039284 s = − 2 − 1 + 2 3 2 + 1 2 1 + 2 3 + i ( 3 2 1 + 2 3 + 3 1 + 2 3 2 ) ≈ − 2.2980358 + 1.8073395 i s = − 2 − 1 + 2 3 2 + 1 2 1 + 2 3 + i ( − 3 1 + 2 3 2 − 3 2 1 + 2 3 ) ≈ − 2.2980358 − 1.8073395 i \begin{aligned}s &= -2 + \sqrt[3]{1 + \sqrt{2}} - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} \approx -1.4039284\\s &= -2 - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right) \approx -2.2980358 + 1.8073395 i\\s &= -2 - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right) \approx -2.2980358 -1.8073395 i\end{aligned} s s s = − 2 + 3 1 + 2 − 3 1 + 2 1 ≈ − 1.4039284 = − 2 − 2 3 1 + 2 + 2 3 1 + 2 1 + i ( 2 3 1 + 2 3 + 2 3 3 1 + 2 ) ≈ − 2.2980358 + 1.8073395 i = − 2 − 2 3 1 + 2 + 2 3 1 + 2 1 + i ( − 2 3 3 1 + 2 − 2 3 1 + 2 3 ) ≈ − 2.2980358 − 1.8073395 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
