Solve x/(x + 100) - (x + 25)/(x + 125) = 0.1.

Step-by-step solution

1. 1

   Note the domain and rewrite $0.1$ as a fraction. The denominators vanish at $x=-100$ and $x=-125$, so those values are excluded from the start. Also write $0.1=\dfrac{1}{10}$ so the whole equation stays in exact arithmetic.

2. 2

   Combine the left side over the common denominator $(x+100)(x+125)$.

   $\frac{x(x+125)-(x+25)(x+100)}{(x+100)(x+125)}$

   Expanding the two products:

   $x(x+125)=x^{2}+125x,\qquad (x+25)(x+100)=x^{2}+125x+2500$

3. 3

   Watch the numerator collapse to a constant. The quadratic and the linear terms cancel completely:

   $\left(x^{2}+125x\right)-\left(x^{2}+125x+2500\right)=-2500$

   so the equation is simply

   $\frac{-2500}{(x+100)(x+125)}=\frac{1}{10}$

   This is the structural heart of the problem: the left side is always negative wherever $(x+100)(x+125)>0$, so the only hope of matching a positive $\tfrac{1}{10}$ is the interval $-125<x<-100$.

4. 4

   Cross-multiply and expand.

   $-25000=(x+100)(x+125)=x^{2}+225x+12500$

   $\Longrightarrow\ x^{2}+225x+37500=0$

5. 5

   Compute the discriminant. With $a=1$, $b=225$, $c=37500$:

   $b^{2}-4ac=50625-150000=-99375<0$

   A negative discriminant means no real roots, so the original equation has no real solution. (Consistent with step 3: on $-125<x<-100$ the product $(x+100)(x+125)$ has maximum magnitude far too small to give $-25000$.)

6. 6

   Report the complex roots if they are wanted. Since $99375=625\cdot 159$, $\sqrt{-99375}=25i\sqrt{159}$ and

   $x=\frac{-225\pm 25i\sqrt{159}}{2}=-112.5\pm 157.619i$

   Substituting either value back into the original expression reproduces $0.1$ to machine precision.