Solve for Vout: 1.2777 + 1.1837 x 0.999Vout x 0.000288 x (284365 + (0.999Vout) squared / 2) - 1.182677 x Vout x 0.000288 x (285105 + Vout squared / 2) = 0.

Step-by-step solution

1. 1

   See the structure: it is a cubic in Vout. Each bracket contributes a constant term and a $V^2$ term, and each is multiplied by $V$, so the equation has the form $A + BV + CV^3 = 0$ with no $V^2$ term. There is no need for a general cubic formula - the numbers decide everything.

2. 2

   Collect the first group's coefficients exactly. $1.1837 \times 0.999 \times 0.000288 = 0.0003405646944$ (not $0.00034074$ - carry all digits). Multiplying by the bracket: the linear part is $0.0003405646944 \times 284365 = 96.844679323$ and the cubic part is $0.0003405646944 \times \tfrac{0.999^2}{2} = 0.000169941953$.

3. 3

   Collect the second group's coefficients. $1.182677\times0.000288 = 0.000340610976$, giving a linear part $0.000340610976\times285105 = 97.109892312$ and a cubic part $\tfrac{0.000340610976}{2} = 0.000170305488$.

4. 4

   Subtract - and respect the cancellation. $B = 96.844679323-97.109892312 = -0.265212989,\qquad C = 0.000169941953-0.000170305488 = -3.635352\times10^{-7}.$ Both differences are catastrophic cancellations: $B$ loses three significant digits and $C$ loses four, so rounding either factor to six digits would destroy the answer. Working with exact fractions is the only safe route here.

5. 5

   Solve the resulting cubic. $1.2777-0.265212989\,V-3.635352\times10^{-7}V^3 = 0.$ The cubic term is negligible near the root, so the linear estimate is $V\approx\tfrac{1.2777}{0.265213}=4.81764$; one Newton correction (or bisection on $[0,20]$) gives $V_{out} \approx 4.81748.$

6. 6

   Verify and note the sensitivity. At $V=4.81748$: $-0.265212989(4.81748) = -1.277658$ and $-3.635352\times10^{-7}(4.81748)^3 = -0.0000406$, and $1.2777-1.277658-0.0000406 \approx 0$. Rounding the first product to $0.00034074$ - the very slip flagged in step 1 - would make $B=-0.215362$ and return $V\approx5.9328$, a $23\%$ error, which is exactly why the cancellation must be handled exactly.