Solve x^3 - 245.711x^2 + 2.764156x + 0.138128 = 0.

Step-by-step solution

1. 1

   Separate the scales first. In $x^{3}-245.711x^{2}+2.764156x+0.138128=0$ Vieta gives root sum $245.711$ and root product $-0.138128$. A large sum with a near-zero product means one dominant root plus two small ones, and the standard trick is to drop whichever term is negligible in each regime.

2. 2

   Find the two small roots by discarding $x^{3}$. If $|x|$ is only a few hundredths then $x^{3}$ is of order $10^{-5}$, utterly negligible beside $245.711x^{2}$. Dropping it leaves the quadratic $245.711x^{2}-2.764156x-0.138128=0,$ whose discriminant is $D=(-2.764156)^{2}+4(245.711)(0.138128)=143.3988344243,\qquad \sqrt{D}=11.9749252367.$ The quadratic formula then gives $x\approx -0.018743095\quad\text{and}\quad x\approx 0.029992718.$

3. 3

   Refine those two with one Newton step. The discarded $x^{3}$ term shifts each root in the seventh decimal place, and applying Newton's method $x\mapsto x-\dfrac{f(x)}{f'(x)}$ to the full cubic converges to $x=-0.018742545\qquad\text{and}\qquad x=0.029994971.$ Compared with the quadratic estimates the corrections are only $5.5\times 10^{-7}$ and $2.3\times 10^{-6}$, which is exactly what "drop $x^{3}$" promised: the neglected term was of size $10^{-5}$ against a $245.711x^{2}$ term of size $10^{-1}$.

4. 4

   Get the large root from the root sum, not from the cubic formula. Since the three roots must add to $245.711$, $x_{\text{large}}=245.711-\left(-0.018742545+0.029994971\right)=245.699747574.$ Substituting this value into $f(x)=x^{3}-245.711x^{2}+2.764156x+0.138128$ returns zero to twelve significant figures.

5. 5

   Verify with both Vieta relations. The three roots $-0.018742545$, $0.029994971$ and $245.699747574$ sum to exactly $245.711000000$ and multiply to $-0.138128$, matching $-c$ as required. The product check is the one that matters here: a pair of small roots can be wrong and still satisfy the sum, because the large root simply absorbs the difference - only the product exposes the error.

6. 6

   State the solution set. $x=-0.018742545,\qquad x=0.029994971,\qquad x=245.699747574.$ All three roots are real, consistent with a cubic that crosses the axis once just below zero, once just above, and once far out at $245.7$.