Solve x^2((8.9 - x)^2 - 64) = (63.9 - 71x)^2.

Step-by-step solution

1. 1

   Resist expanding, and factor the difference of squares on the left. Since $64=8^{2}$, $(8.9-x)^{2}-64=(8.9-x-8)(8.9-x+8)=(0.9-x)(16.9-x),$ so the left side is $x^{2}(0.9-x)(16.9-x)$. Multiplying everything out would produce a general quartic and hide the structure that makes this problem solvable by hand.

2. 2

   Factor the right side too. Notice $71\times 0.9=63.9$, so $63.9-71x=71(0.9-x)\quad\Longrightarrow\quad (63.9-71x)^{2}=71^{2}(0.9-x)^{2}=5041(0.9-x)^{2}.$ Spotting that $63.9$ is $71\times 0.9$ is the key observation of the whole problem.

3. 3

   Move everything to one side and pull out $(0.9-x)$. $x^{2}(0.9-x)(16.9-x)-5041(0.9-x)^{2}=0$ $(0.9-x)\left[x^{2}(16.9-x)-5041(0.9-x)\right]=0$ The quartic has now split into a linear factor and a cubic, so a degree-4 equation becomes something we can actually finish.

4. 4

   Read off the exact root. The first factor gives $0.9-x=0\quad\Longrightarrow\quad x=0.9,$ and substituting back confirms it: the left side becomes $0.81(64-64)=0$ and the right side becomes $(63.9-63.9)^{2}=0$.

5. 5

   Solve the remaining cubic. Expanding the bracket, $16.9x^{2}-x^{3}+5041x-4536.9=0\quad\Longleftrightarrow\quad x^{3}-16.9x^{2}-5041x+4536.9=0,$ or with decimals cleared, $10x^{3}-169x^{2}-50410x+45369=0$. It has no rational roots, and numerically its three real roots are $x\approx -63.548577,\qquad x\approx 0.897443,\qquad x\approx 79.551134.$

6. 6

   Check the cubic roots with Vieta's relations. For $x^{3}-16.9x^{2}-5041x+4536.9$ the roots must sum to $16.9$, have pairwise-product sum $-5041$, and multiply to $-4536.9$. The three numbers give $-63.548577+0.897443+79.551134=16.900000$, pairwise sum $-5041.000000$, and product $-4536.899998$ - all three match. Note how close $0.897443$ is to the exact root $0.9$: they differ by only $0.0026$, so a rounded solution can easily be mistaken for the exact one.

7. 7

   List the full solution set. $x=0.9,\qquad x\approx -63.548577,\qquad x\approx 0.897443,\qquad x\approx 79.551134.$ All four are real, which is consistent with a quartic whose leading coefficient is positive on the difference side and which changes sign four times.