# solve equation of form $x = a - (b - x)^c$

Is there a general solution (for $x$) to an equation of the form $x = a - (b - x)^c$ ? If there is no general solution, please suggest a strategy for getting a numerical answer. In the case I care about immediately, two different computer algebra systems got as far as $x = 50\,000 - (100 - x)^{500\,000}$ and then got stuck when asked for a numeric answer.

• If you let $b-x=y$, you get a nicer equation $$b - a = y - {y^c}$$
– Pedro
Nov 7, 2012 at 0:07

Assuming $a,b,x$ are reals and $c$ is an integer, your equation becomes

$$a-(b-x)^c-x = 0$$

which can be converted to

$$a-x-\sum_{i=0}^c \begin{pmatrix} c \\ i \end{pmatrix} b^ix^{c-i} = 0$$

using the binomial theorem.

For $c > 4$, there is no general solution for this polynomial by the Abel-Ruffini theorem. However, it may be possible that for certain specific conditions on $a, b \in \Bbb R$ that allow it to be solved.

With Mathematica

FindRoot[50000 - (100 - x)^500000 == x, {x, 99}, AccuracyGoal -> 20,
WorkingPrecision -> 50]


I get x = 98.999978364173304772436016428458876920765894232268 You can get more accurate values by increasing the WorkingPrecision. You can use Newton's Method with something like this.

If such an x exists it will be between 98 and 99, and by between them I mean it's going to be very close to 99.

Using maple I could maybe get close just by messing around with it. If such an x exists it is between x = 98.9999783650000000000000000000000000000000000000000000001 and x = 98.9999783650000000000000000000000000000000000000000000000

The fact that maple chokes on it tells me that an x likely does not exist. I used this expression to look for it: limit(ln(50000-x)/ln(100-x), x = 98.9999783650000000000000000000000000000000000000000000000, left)