Solve for x: 8/(8.9 - x) = (71 - x times root((8.9 - x)^2 - 64)/(8.9 - x))/71.

Step-by-step solution

1. 1

   Fix the domain first. We need $8.9-x\neq0$ and $(8.9-x)^2-64\ge0$, i.e. $|8.9-x|\ge8$, which means

   $x\le 0.9\qquad\text{or}\qquad x\ge 16.9$

   Every candidate root must survive this test, and squaring later will manufacture roots that do not.

2. 2

   Clear both denominators. Multiplying by $71$ and then by $8.9-x$:

   $568=71(8.9-x)-x\sqrt{(8.9-x)^2-64}$

   Since $71(8.9)=631.9$, this rearranges to

   $x\sqrt{(8.9-x)^2-64}=631.9-71x-568=63.9-71x$

3. 3

   Factor the radicand as a difference of squares. This is the step that makes the problem tractable:

   $(8.9-x)^2-64=(8.9-x-8)(8.9-x+8)=(0.9-x)(16.9-x)$

   and at the same time $63.9-71x=71(0.9-x)$. The same factor $0.9-x$ appears on both sides, so the equation is

   $x\sqrt{(0.9-x)(16.9-x)}=71(0.9-x)$

4. 4

   Read off the first solution for free. If $0.9-x=0$ both sides are zero, so $x=0.9$ is an exact solution. Checking in the original: $\tfrac{8}{8}=1$ on the left, and on the right the radical is $0$, giving $\tfrac{71-0}{71}=1$ ✓.

5. 5

   Handle the remaining branch. For $x<0.9$ we have $0.9-x>0$, so $\sqrt{(0.9-x)(16.9-x)}=\sqrt{0.9-x}\,\sqrt{16.9-x}$ and one factor of $\sqrt{0.9-x}$ cancels:

   $x\sqrt{16.9-x}=71\sqrt{0.9-x}\ \Longrightarrow\ x^{2}(16.9-x)=5041(0.9-x)$

   $\Longrightarrow\ x^{3}-16.9x^{2}-5041x+4536.9=0$

   (The branch $x\ge16.9$ is impossible: there the left side is $\ge0$ while $71(0.9-x)<0$.)

6. 6

   Solve the cubic and reject the extraneous roots. Its roots are $-63.5486$, $0.8974433$ and $79.5511$. Squaring required $x>0$ and $x\le0.9$, so the first and third are extraneous - substituting them into the original equation gives $-1.78$ and $-2.23$ instead of $0$. Only

   $x=0.8974433$

   survives, and it reproduces the original equation to $10^{-16}$ ✓.

7. 7

   State both answers. The full solution set is $x=0.9$ and $x\approx0.8974433$. Note $(x-0.9)$ times the cubic reproduces the quartic $x^4-17.8x^3-5025.79x^2+9073.8x-4083.21$ exactly, confirming no root was lost.