Solve x4 + x3 + x2 + x = 4553.

Step-by-step solution

1. 1

   Factor the left side to learn the shape. Grouping,

   $x^4+x^3+x^2+x=x(x^3+x^2+x+1)=x(x+1)(x^2+1)$

   Because $x^2+1>0$ always, the sign of $g(x)=x(x+1)(x^2+1)$ is the sign of $x(x+1)$: negative only on $(-1,0)$, positive outside. And $g$ grows like $x^4$ in both directions. So for any target above $0$ there are exactly two real roots, one positive and one negative — the negative one is the half that gets forgotten.

2. 2

   Bracket the positive root. $g(7)=2401+343+49+7=2800$ and $g(8)=4096+512+64+8=4680$. The target $4553$ lies between them, so the root is in $(7,8)$ and close to $8$.

3. 3

   Refine the positive root. $g(7.9)=4458.36$ and $g(7.95)=4568.17$, so the root is in $(7.9,7.95)$. Bisecting to machine precision gives

   $x_1=7.9431476$

4. 4

   Reject the commonly quoted 7.949102. Evaluating there, $g(7.949102)=4566.18$, which overshoots $4553$ by more than $13$. A value that far off cannot be the root; the discrepancy is easy to miss because the fourth power makes $g$ change very fast near $x=8$ (about $2200$ per unit).

5. 5

   Bracket and refine the negative root. $g(-8)=4096-512+64-8=3640$ and $g(-8.5)=4669.69$, so the second root lies in $(-8.5,-8)$. Bisection gives

   $x_2=-8.4477787$

6. 6

   Verify both. $g(7.9431476)=4553.000$ and $g(-8.4477787)=4553.000$ to three decimals. The remaining two roots of the quartic are a complex-conjugate pair, so the real solution set is exactly $\{7.9431476,\,-8.4477787\}$.