Roots lying between the roots of a given equation

[utkarshakash]

Homework Statement

For what real values of 'a' do the roots of the equation $x^2-2x-(a^2-1)=0$ lie between the roots of the equation $x^2-2(a+1)x+a(a-1)=0$

Homework Equations

The Attempt at a Solution

The required conditions are
$\large D_1\geq0$
$\large D_2\geq0$
$\large R_1<\dfrac{-2}{2}<R_2$
where
$D_n=$Discriminant of nth equation and $R_m=$ mth roots of the latter equation

Solving simultaneously the above inequalities and taking intersection
$a\in \left(\frac{-1}{3},0\right)$
But the correct answer is
$a\in \left(\frac{-1}{4},1\right)$

 

[haruspex]

Where does the -2/2 come from? Shouldn't there be an 'a' in there?

 

[utkarshakash]

Where does the -2/2 come from? Shouldn't there be an 'a' in there?

Sorry I made a mistake. It must be 2/2 (-b/2a)

 

[haruspex]

Sorry I made a mistake. It must be 2/2 (-b/2a)

But that's only the average of the roots. You need the condition that each root individually is in that range.

 

[utkarshakash]

But that's only the average of the roots. You need the condition that each root individually is in that range.

You are not understanding. Its not the average of the roots. Its the x-coordinate at which the minimum value of function occurs.

 

[haruspex]

You are not understanding. Its not the average of the roots. Its the x-coordinate at which the minimum value of function occurs.

OK, but why is that interesting?

 

[ehild]

Express the roots in terms of 'a' first. The non-negativity of the discriminants is not enough.

If you are right, both roots of the first equation lie between or equal to one of the roots of the second equation in case a=-1/3. Is it true?

ehild

 

[utkarshakash]

Express the roots in terms of 'a' first. The non-negativity of the discriminants is not enough.

If you are right, both roots of the first equation lie between or equal to one of the roots of the second equation in case a=-1/3. Is it true?

ehild

I started by finding roots then substituting them for x in the latter equation and setting them <0. The answer which I get is correct. So, no worries!

 

[ehild]

Did you get $a\in \left(\frac{-1}{4},1\right)$? That is the correct answer, I have checked. Why should the roots be negative? Try with a=0.5. Do the roots of the first equation lie between the roots of the second one?


ehild

 

[utkarshakash]

Yes i got that answer.