Solve x^3 - 245.711x^2 + 2.764156x + 0.115212 = 0.

Step-by-step solution

1. 1

   Use the lopsided coefficients to predict the root pattern. With $x^{3}-245.711x^{2}+2.764156x+0.115212=0$ the sum of the roots is $245.711$ and their product is $-0.115212$. Those two facts cannot both hold unless one root is around $245.7$ and the other two are within a few hundredths of zero - so treat the two regimes separately instead of reaching for the general cubic formula.

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 $245.711x^{2}-2.764156x-0.115212=0,$ whose discriminant is $D=(-2.764156)^{2}+4(245.711)(0.115212)=120.8759813203,\qquad \sqrt{D}=10.9943613421.$ The quadratic formula then gives $x\approx -0.016747735\quad\text{and}\quad x\approx 0.027997357.$

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 $x=-0.016747308\qquad\text{and}\qquad x=0.027999354.$ Compared with the quadratic estimates the corrections are only $4.3\times 10^{-7}$ and $2.0\times 10^{-6}$, 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 $245.711$, $x_{\text{large}}=245.711-\left(-0.016747308+0.027999354\right)=245.699747954.$ Substituting this value into $f(x)=x^{3}-245.711x^{2}+2.764156x+0.115212$ returns zero to twelve significant figures.

5. 5

   Verify with both Vieta relations. The three roots $-0.016747308$, $0.027999354$ and $245.699747954$ sum to exactly $245.711000000$ and multiply to $-0.115212$, 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. $x=-0.016747308,\qquad x=0.027999354,\qquad x=245.699747954.$ 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 $245.7$.