Solve x^3 - 245.711x^2 + 2.764156x + 0.093547 = 0.

Step-by-step solution

1. 1

   Read the coefficient sizes before doing any algebra. For $x^{3}-245.711x^{2}+2.764156x+0.093547=0$, Vieta's relations give a root sum of $245.711$ and a root product of $-0.093547$. A sum in the hundreds paired with a product of a few hundredths forces one large root and two roots near zero, and each group needs its own approximation.

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 $245.711x^{2}-2.764156x-0.093547=0,$ whose discriminant is $D=(-2.764156)^{2}+4(245.711)(0.093547)=99.5826660603,\qquad \sqrt{D}=9.9791114865.$ The quadratic formula then gives $x\approx -0.014681792\quad\text{and}\quad x\approx 0.025931414.$

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 $x=-0.014681475\qquad\text{and}\qquad x=0.025933162.$ Compared with the quadratic estimates the corrections are only $3.2\times 10^{-7}$ and $1.7\times 10^{-6}$, 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 $245.711$, $x_{\text{large}}=245.711-\left(-0.014681475+0.025933162\right)=245.699748313.$ Substituting this value into $f(x)=x^{3}-245.711x^{2}+2.764156x+0.093547$ returns zero to twelve significant figures.

5. 5

   Verify with both Vieta relations. The three roots $-0.014681475$, $0.025933162$ and $245.699748313$ sum to exactly $245.711000000$ and multiply to $-0.093547$, 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. $x=-0.014681475,\qquad x=0.025933162,\qquad x=245.699748313.$ 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 $245.7$.