Solve 71 × 8 = 71(8.9 − x) − x · sqrt((8.9 − x)² − 64) for x.

Step-by-step solution

1. 1

   Substitute $u=8.9-x$ so the radicand factors. The expression $(8.9-x)^2-64$ is a difference of squares in $u$. With $u=8.9-x$ (so $x=8.9-u$ and $71\cdot 8=568$) the equation becomes

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

   $71(u-8)=(8.9-u)\sqrt{(u-8)(u+8)}$

   Writing $568$ as $71\cdot 8$ is what makes the left side factor — that is the whole reason this equation has a clean root.

2. 2

   Fix the domain and kill one branch immediately. The radical needs $u^2\ge 64$, i.e. $u\ge 8$ or $u\le -8$.

   If $u\le -8$, the left side $71(u-8)$ is negative while the right side $(8.9-u)\sqrt{u^2-64}$ has a positive factor $8.9-u>0$ times a non-negative root, so the right side is $\ge 0$. No solution there — which also rules out every $x\ge 16.9$.

   So $u\ge 8$, i.e. $x\le 0.9$. On that branch the left side is $\ge 0$, so we also need $x=8.9-u\ge 0$: the search narrows to $0\le x\le 0.9$.

3. 3

   Set $w=u-8\ge 0$ and read off the boundary root. Then $8.9-u=0.9-w$ and

   $71w=(0.9-w)\sqrt{w(w+16)}$

   Taking $w=0$ satisfies this exactly: both sides are $0$. That gives $u=8$ and

   $x=8.9-8=0.9$

   Check it in the original equation: $71(8.9-0.9)-0.9\sqrt{8^2-64}=568-0=568$. ✓ This is the root that a purely numerical solver most often misses, because it sits exactly on the edge of the domain where the square root turns on.

4. 4

   Handle $w>0$ by squaring, then divide out the factor $w$. Squaring gives $5041w^2=(0.9-w)^2w(w+16)$, and dividing by $w>0$:

   $5041w=(w^2-1.8w+0.81)(w+16)=w^3+14.2w^2-27.99w+12.96$

   $w^3+14.2w^2-5068.99w+12.96=0$

   Its three roots are $w\approx -78.6511$, $w\approx 0.00255674$ and $w\approx 64.4486$. Only the tiny positive one satisfies $0<w<0.9$; the other two make $0.9-w$ negative while $71w$ is positive (or vice versa), so squaring introduced them.

5. 5

   Convert back and verify both answers numerically. From $x=0.9-w$:

   $x=0.9-0.0025567407\ldots=0.8974432593\ldots$

   Substituting into the original equation gives $71(8.0025567)-0.8974433\sqrt{8.0025567^2-64}=568.0000000$, residual $0$ to machine precision. Together with the boundary root:

   $x=0.9\qquad\text{and}\qquad x\approx 0.8974432593$

6. 6

   Why the popular shortcut answer $x\approx 0.797$ is wrong. A common route squares first and then tries to factor the quartic $x^4-17.8x^3-5025.79x^2+9073.8x-4083.21$ as $(x^2+63.2x-63.9)(x^2-81x+63.9)$. Expanding that product gives an $x^2$ coefficient of $-5119.2$, not $-5025.79$, so the factorization is simply false and the values $0.7968$ and $0.9954$ it produces are not roots. Substituting $x=0.7968$ into the original equation returns about $574.30$, not $568$; and the companion value $0.9954$ is not even in the domain, since $(8.9-0.9954)^2-64\approx -1.52<0$.