I am having difficulty with understanding one part of the answer to the following:
Find conditions on a, b, and c which are necessary and sufficient to ensure that the equations $$ a\cosh x + b\sinh x = c $$ has exactly one root.
I have rearranged the equation to form a quadratic equation and used the discriminate and got:
$$ a^2 = c^2 + b^2 $$
However, the answer is that the above statement is only true if the signs of a+b and a-b are the same. When the signs of a+b and a-b are different then
$$ b^2+c^2>a^2 $$
This is the part that I don't understand how to prove. I have used graphing software to see that it is true but I can't seem to prove it. Could you help with this please?
If you let $e^x=y$, you end with the equation $$ (a+b)y^2-2 c y+(a-b)=0$$ So, first we need two roots $$\Delta=b^2+c^2-a^2 \geq 0$$ and one of them must be negative since $y >0$ so $\frac{a-b}{a+b} <0$ that is to say $a^2-b^2 <0$