Proving 0 < θ1 < β in θ12-γθ1+β=0

[Kinetica]

Homework Statement

θ₁²-γθ₁+β=0

Show that

0<θ₁<β

The Attempt at a Solution

I know that for θ₁=0, θ₁²-γθ₁+β>0:
Substituting, we get 0²-0+β=β, which is positive.

I don't know how to show that for θ₁=λ, θ₁²-γθ₁+β<0.

I also don't know how to show that these results imply that there is zero between these two values. Which in turns means that 0<θ₁<β.

 

[Joffan]

Well, θ₁²-γθ₁+β=0 is just a quadratic equation in θ₁, so there should be at most two possible values for θ₁, in terms of the other parameters.

$$\theta_{1} = \frac{\gamma\pm\sqrt{\gamma^{2}-4\beta}}{2}$$

But as things stand, you haven't given nearly enough information to assert the inequalities required. For example, why shouldn't β be zero or even negative? Is γ greater than zero, greater than one?

 

[conquest]

consider for instance β=0 and y=θ then θ can be whatever you like as long as y is also!

 

[Joffan]

OK so I found this on your other request, and it looks like it could apply here.

θ₁+θ₂=γ
θ₁*θ₂=β

but that just means that the relative sizes of θ₁ and β depend on θ₂. So still no closer to that inequality - there's something you aren't telling us about this question...

 

[SammyS]

Homework Statement

θ₁²-γθ₁+β=0

Show that

0<θ₁<β

The Attempt at a Solution

I know that for θ₁=0, θ₁²-γθ₁+β>0:
Substituting, we get 0²-0+β=β, which is positive.

I don't know how to show that for θ₁=λ, θ₁²-γθ₁+β<0.

I also don't know how to show that these results imply that there is zero between these two values. Which in turns means that 0<θ₁<β.

This thread is very similar in topic to a thread you started one day earlier: https://www.physicsforums.com/showthread.php?p=3717073#post3717073 .

Where did λ (lambda) come from, or should that be γ (gamma) ?

I suppose we can infer that β > 0 from the inequality, 0<θ₁<β, and because you mentioned it in passing, "Substituting, we get 0²-0+β=β, which is positive."

It makes no sense to plug values such as 0 or γ or λ in for θ₁ in the quadratic polynomial θ₁²-γθ₁+β to see if 0<θ₁<β .

As Joffan said, you need more information regarding β and γ, before you can say much about θ₁ .