The following equation is given $$\log_{x-1}(x^2+2ax) - \log_{x-1}(8x-6a-3)=0$$
And I am trying to find for which values of $a$ it has only one root, which is real.
It is obvious that $$x-1>0 \Rightarrow x>1, x\neq2 $$
Also $$x^2+2ax>0$$ $$8x-6a-3>0 \Rightarrow x>\frac{6a+3}8 $$
Now I combine the two logarithms $$\log_{x-1}\left(\frac{x(x+2a)}{8x-6a-3}\right)=0$$ and then $$\frac{x(x+2a)}{8x-6a-3}=1$$ $$\frac{x^2+2(a-4)x+6a+3}{8x-6a-3}=0$$ $$x^2+2(a-4)x+6a+3=0$$ The discriminant is $$D=(a-13)(a-1)$$ and the roots are $$x_1= -a+4-\sqrt{(a-13)(a-1)}$$ $$x_2= -a+4+\sqrt{(a-13)(a-1)}$$ so that $x_1<x_2$.
- Case 1:
For $D=0$ we have only one root for $x$. $D=0$ for $a=13$ or $a=1$.
For $a=13$, $x_1=x_2=-9\not\gt1$ so $a=13$ is not a solution.
For $a=1$, $x_1=x_2=3>1$. Also $x=3>\frac{6a+3}{8}=\frac98$ and $x^2+2ax=15>0$, so for $a=1$ we have only one root, therefore it is a solution.
- Case 2:
Now let $D>0$, so that $x_1 \ne x_2$. $D>0$ for $a\in(-\infty,1)\cup(13,+\infty)$
I consider five sub-cases:
A) When $x_1 = 2$ and $x_2^2+2ax_2>0, x_2>\frac{6a+3}8 $
B) When $x_2=2$ and $x_1>1, x_1^2+2ax_1>0, x_1>\frac{6a+3}8 $.
C) When $1\in[x_1,x_2)$ and $x_2^2+2ax_2>0, x_2>\frac{6a+3}8 $.
D) When $\frac{6a+3}8\in[x_1,x_2)$ and $x_2>1, x_2^2+2ax_2>0 $.
E) When $x_1^2+2ax_1\le0$ and $x_2>1, x_2^2+2ax_2>0, x_2>\frac{6a+3}8 $
The solutions I find are
A) $a=\frac9{10}, x_2=\frac{21}5$
B) none
C), D), E) It all comes up to solving inequalities involving polynomials for $a$ of degree 4, which makes me doubt I am doing it right.
Am I missing any cases or am I considering too many cases? How would you approach this problem?