Solve 0.2 = 4.6092x^3 + 9.2964x^2 + 2.7507x + 0.0251 for x.

Step-by-step solution

1. 1

   Move everything to one side. Subtracting $0.2$:

   $f(x)=4.6092x^{3}+9.2964x^{2}+2.7507x-0.1749=0$

   A root of $f$ is what we want. The coefficients are all positive except the constant, so by Descartes rule there is exactly one positive real root.

2. 2

   Bracket the root with an interval where the sign changes. Evaluate carefully:

   $f(0)=-0.1749,\qquad f(0.05)=-0.0135,\qquad f(0.06)=+0.0246$

   The sign flips between $0.05$ and $0.06$, so the root lies there. Every digit of these test values matters — a mis-evaluated $f(0.06)$ would send the bracket to the wrong interval and produce a confidently wrong answer.

3. 3

   Check the arithmetic of the decisive test point. At $x=0.06$: $4.6092(0.000216)=0.000996$, $9.2964(0.0036)=0.033467$, $2.7507(0.06)=0.165042$. Summing and subtracting: $0.199505-0.1749=+0.024605$. Positive, confirming the bracket $[0.05,0.06]$.

4. 4

   Refine with Newton method. With $f'(x)=13.8276x^{2}+18.5928x+2.7507$ and $x_{0}=0.055$, successive iterates are $0.053626$, $0.0536118$, $0.0536118$ — converged to seven decimals:

   $x\approx 0.0536118$

5. 5

   Verify the root and account for the others. $f(0.0536118)=0.0000000$ ✓. The other two roots of the cubic, $x\approx-0.43195$ and $x\approx-1.63858$, are negative; if the variable represents a physical quantity they are usually discarded, leaving $x\approx 0.0536$ as the answer.