Hello, Any idea why I obtain a huge numerical precision using complexSolve here, at least, displayed?
(1) -> complexSolve(x^3+3*x^2+x+17=0, 0.0000000001)
(1)
[x = - 3.8744001380_296332337,
x = 0.4372000690_1481618814_9406395091_6108308765_2840423528 -
2.0485683086_313182244 %i,
x
=
0.4372000690_1481618814_9406395091_6108308765_2840423528
+
2.0485683086_3131822439_9819882819_4290399551_391601562 %i
]
Type:
List(Equation(Polynomial(Complex(Float))))
(2) -> precision()$Float
(2) 68
Type: PositiveInteger
(3) -> digits()$Float
(3) 20
Regards,
Greg
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Title: Cubic Equation Solver - Results
Cubic Equation Solved
Equation: \(x^3 + 3x^2 + x + 17 = 0\)
\[x^3 + 3x^2 + x + 17 = 0\]
First Root (\(x_1\))
Real Root
\[x_1 = \frac{\left(-3\sqrt{-3}+3\right)\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}+\left(-3\sqrt{-3}-3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}-4}{\left(3\sqrt{-3}+3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}\]
x_1 = \frac{\left(-3\sqrt{-3}+3\right)\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}+\left(-3\sqrt{-3}-3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}-4}{\left(3\sqrt{-3}+3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}
Second Root (\(x_2\))
Complex Conjugate
\[x_2 = \frac{\left(-3\sqrt{-3}-3\right)\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}+\left(-3\sqrt{-3}+3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}+4}{\left(3\sqrt{-3}-3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}\]
x_2 = \frac{\left(-3\sqrt{-3}-3\right)\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}+\left(-3\sqrt{-3}+3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}+4}{\left(3\sqrt{-3}-3\right)\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}
Third Root (\(x_3\))
Complex Conjugate
\[x_3 = \frac{3\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}-3\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}+2}{3\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}\]
x_3 = \frac{3\left(\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}\right)^{2}-3\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}+2}{3\sqrt[3]{\frac{\sqrt{\frac{2179}{3}}-27}{3}}}
Real Root
x1
x = -3.8744001380296332337...
Complex Conjugate Root 1
x2
x = 0.43720006901481618814940639509161083087652840423528 - 2.0485683086313182244 i
Complex Conjugate Root 2
x3
x = 0.43720006901481618814940639509161083087652840423528 + 2.0485683086313182244 i
Step 1: Depress the Cubic
Substitute \(x = y - 1\) to eliminate the quadratic (\(x^2\)) term. The equation becomes:
\[y^3 - 2y + 18 = 0\]Step 2: Apply Cardano's Formula
For a depressed cubic of the form \(y^3 + py + q = 0\), the solution is given by \(y = u + v\), where:
\[u^3 + v^3 = -q = -18, \quad u \cdot v = -\frac{p}{3} = \frac{2}{3}\]Solving the quadratic equation for \(u^3\) gives:
\[u = \sqrt[3]{-9 + \sqrt{\frac{2179}{27}}} \approx -0.25445209\] \[v = \frac{2}{3u} \approx -2.61994805\]Step 3: Construct the Three Roots
Using the primitive cube root of unity \(\omega = \frac{-1 + i\sqrt{3}}{2}\), we obtain the values for \(x = y - 1\):
\[x_1 = u + v - 1 \approx -3.874400138\] \[x_2 = \omega u + \omega^2 v - 1 \approx 0.437200069 - 2.048568309 i\] \[x_3 = \omega^2 u + \omega v - 1 \approx 0.437200069 + 2.048568309 i\]