Solve the cubic equation x^3 + 40x^2 + 400x + 2000 = 0.

Step-by-step solution

1. 1

   Check the rational candidates. Divisors of $2000$ include $\pm 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 400, 500, 1000, 2000$. Evaluating the plausible negatives gives $f(-10) = 1000$, $f(-20) = 2000$, $f(-25) = -1375$, $f(-40) = -14000$: no zeros, but the sign flip between $-20$ and $-25$ pins one real root in that range.

2. 2

   Depress the cubic. With $a = 40$, $b = 400$, $c = 2000$, substitute $x = y - \tfrac{40}{3}$. Then

   $p = b - \frac{a^2}{3} = 400 - \frac{1600}{3} = -\frac{400}{3}, \qquad q = \frac{2a^3}{27} - \frac{ab}{3} + c = \frac{128000}{27} - \frac{144000}{27} + \frac{54000}{27} = \frac{38000}{27}$

   giving $y^3 - \tfrac{400}{3}y + \tfrac{38000}{27} = 0$.

3. 3

   Compute the discriminant.

   $\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \left(\frac{19000}{27}\right)^2 - \left(\frac{400}{9}\right)^3 = \frac{361000000 - 64000000}{729} = \frac{11000000}{27}$

   That is $\Delta \approx 407407.4 > 0$, so one real root and two complex conjugates.

4. 4

   Apply Cardano and shift back.

   $x = \sqrt[3]{-\frac{19000}{27} + \sqrt{\frac{11000000}{27}}} + \sqrt[3]{-\frac{19000}{27} - \sqrt{\frac{11000000}{27}}} - \frac{40}{3}$

   Numerically $\sqrt{11000000/27} = 638.2847$ and $\tfrac{19000}{27} = 703.7037$, so the cube roots are $\sqrt[3]{-65.4190} = -4.02935$ and $\sqrt[3]{-1341.9884} = -11.03018$. Their sum is $y \approx -15.059534$, hence

   $x_1 = -15.059534 - 13.333333 = -28.392868$

5. 5

   Recover the complex pair. Dividing out $(x - x_1)$ leaves $x^2 + 11.607132x + 70.440226$, whose roots are

   $x = -5.803566 \pm 6.062907\,i$

6. 6

   Verify with Vieta. Sum of roots $= -28.392868 + 2(-5.803566) = -40.000000$, matching $-a = -40$. Product $= -28.392868\left(5.803566^2 + 6.062907^2\right) = -28.392868 \times 70.440220 = -2000.0$, matching $-c = -2000$. Direct substitution of $x_1$ into the cubic returns $|f(x_1)| < 10^{-9}$. (A frequently quoted value of $-30.3393$ for the real root is wrong: it gives $f(-30.3393) \approx -1243$.)