Solve x^3 - 245.711x^2 + 2.764156x + 0.074296 = 0.

Step-by-step solution

1. 1

   See why the roots must split into one huge one and two tiny ones. In $x^{3}-245.711x^{2}+2.764156x+0.074296=0$ the quadratic coefficient dwarfs everything else. Vieta says the three roots sum to $245.711$ and multiply to $-0.074296$, a number close to zero - so one root has to carry almost the whole sum while the other two are small enough to keep the product tiny. Recognising that structure is what lets us split the problem into two easy sub-problems.

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 $245.711x^{2}-2.764156x-0.074296=0,$ whose discriminant is $D=(-2.764156)^{2}+4(245.711)(0.074296)=80.6619362163,\qquad \sqrt{D}=8.9811990411.$ The quadratic formula then gives $x\approx -0.012651129\quad\text{and}\quad x\approx 0.023900751.$

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 $x=-0.012650903\qquad\text{and}\qquad x=0.023902272.$ Compared with the quadratic estimates the corrections are only $2.3\times 10^{-7}$ and $1.5\times 10^{-6}$, 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 $245.711$, $x_{\text{large}}=245.711-\left(-0.012650903+0.023902272\right)=245.699748632.$ Substituting this value into $f(x)=x^{3}-245.711x^{2}+2.764156x+0.074296$ returns zero to twelve significant figures.

5. 5

   Verify with both Vieta relations. The three roots $-0.012650903$, $0.023902272$ and $245.699748632$ sum to exactly $245.711000000$ and multiply to $-0.074296$, 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. $x=-0.012650903,\qquad x=0.023902272,\qquad x=245.699748632.$ 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 $245.7$.