Quadratic Equation: Finding Values of 'a' Between 2 and 4 with Only One Root

[Saitama]

Homework Statement

Find all the values of 'a', so that exactly one root of the equation x²-2ax+a²-1=0, lies between the numbers 2 and 4, and no root is either equal to 2 or equal to 4.

Homework Equations

The Attempt at a Solution

Let f(x)=x²-2ax+a²-1
I tried to visualize the question by graph. The graph could have been like this (this is only a rough sketch):-

From here, i get three inequalities,
f(2)<0 and f(4)>0 and D>0
Solving these inequalities, i get a can lie in interval (1,3).
But this is not the answer, the answer is (1,5)-{3}.
Then i thought that graph could be also like this:-

But this gives completely different set of inequalities, now i am completely stuck.

Any help is appreciated. :)

 

[hqjb]

Homework Statement

Find all the values of 'a', so that exactly one root of the equation x²-2ax+a²-1=0, lies between the numbers 2 and 4, and no root is either equal to 2 or equal to 4.

Homework Equations

The Attempt at a Solution

Let f(x)=x²-2ax+a²-1
I tried to visualize the question by graph. The graph could have been like this (this is only a rough sketch):-

From here, i get three inequalities,
f(2)<0 and f(4)>0 and D>0
Solving these inequalities, i get a can lie in interval (1,3).
But this is not the answer, the answer is (1,5)-{3}.
Then i thought that graph could be also like this:-

But this gives completely different set of inequalities, now i am completely stuck.

Any help is appreciated. :)

The quadratic formula simplifies the root to a function of "a", no need to use graphs, I think your right btw (1,5)-{3} is the union of (1,3) and (3,5)

 

[I like Serena]

Hey Pranav-Arora!

I tried to visualize the question by graph. The graph could have been like this (this is only a rough sketch):-

Aha! You're drawing again. Good!

Solving these inequalities, i get a can lie in interval (1,3).

Yep.

Then i thought that graph could be also like this:-
But this gives completely different set of inequalities, now i am completely stuck.

Not stuck.
This is indeed another set of inequalities that you also have to solve.
The solution is the combination of both solutions.

 

[Saitama]

The quadratic formula simplifies the root to a function of "a", no need to use graphs, I think your right btw (1,5)-{3} is the union of (1,3) and (3,5)

Oh yes, i knew about it, it will reduce to (x-a)²-1=0 and then we can proceed on the next steps. Thanks for the reply! :)

Hello ILS!

Aha! You're drawing again. Good!

Thanks.

Not stuck.
This is indeed another set of inequalities that you also have to solve.
The solution is the combination of both solutions.

Now i understand it. Thanks once again!

 

[stef6987]

find the square and solve :D