Solve the cubic equation 2s^3 + 33s^2 + 180s + 300 = 0.

Step-by-step solution

1. 1

   Rule out rational roots. Candidates are $\pm\tfrac{d}{e}$ with $d \mid 300$ and $e \mid 2$. Testing the plausible negatives: $f(-3) = 3$, $f(-4) = -20$, $f(-5) = -25$, $f(-6) = -24$, $f(-10) = -200$, $f(-15) = -1725$. None vanish, so factoring and synthetic division are both dead ends — the sign change between $-3$ and $-4$ does tell us one real root lies in that gap.

2. 2

   Make the cubic monic and depress it. Dividing by $2$ gives $s^3 + \tfrac{33}{2}s^2 + 90s + 150 = 0$. The substitution $s = y - \tfrac{a}{3}$ with $a = \tfrac{33}{2}$, i.e. $s = y - \tfrac{11}{2}$, removes the square term. Using $p = b - \tfrac{a^2}{3}$ and $q = \tfrac{2a^3}{27} - \tfrac{ab}{3} + c$:

   $p = 90 - \frac{1089}{12} = -\frac{3}{4}, \qquad q = \frac{1331}{4} - 345 = -\frac{49}{4}$

   so the depressed cubic is $y^3 - \tfrac{3}{4}y - \tfrac{49}{4} = 0$.

3. 3

   Compute the discriminant to see how many real roots there are.

   $\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \frac{2401}{64} - \frac{1}{64} = \frac{2400}{64} = \frac{75}{2} = 37.5$

   Since $\Delta > 0$ there is exactly one real root and a conjugate pair of complex roots. This is the case where Cardano's radical formula works with real cube roots — no trigonometric detour needed.

4. 4

   Apply Cardano's formula. With $-\tfrac{q}{2} = \tfrac{49}{8} = 6.125$ and $\sqrt{\Delta} = \sqrt{37.5} = 6.123724$:

   $y = \sqrt[3]{6.125 + 6.123724} + \sqrt[3]{6.125 - 6.123724} = \sqrt[3]{12.248724} + \sqrt[3]{0.001276}$

   The two radicands are wildly different in size — the second is barely above zero because $\Delta$ falls just short of $\left(\tfrac{q}{2}\right)^2$ — giving

   $y = 2.305138 + 0.108453 = 2.413591$

5. 5

   Shift back to s and find the complex pair. With $s = y - \tfrac{11}{2}$ the real root is

   $s_1 = 2.413591 - 5.5 = -3.086409$

   Dividing $2s^3 + 33s^2 + 180s + 300$ by $(s - s_1)$ leaves $2s^2 + 26.827182s + 97.200344$, whose roots are $s = -6.706796 \pm 1.902385\,i$.

6. 6

   Check with Vieta. The three roots must sum to $-\tfrac{33}{2} = -16.5$: $-3.086409 + 2(-6.706796) = -16.500001$. Their product must be $-\tfrac{300}{2} = -150$: $(-3.086409)\left(6.706796^2 + 1.902385^2\right) = (-3.086409)(48.60018) = -150.00$. Both match. (A commonly seen but wrong answer gives the imaginary part as $\pm 0.484891$; that pair fails the product test badly.)