How to approach this problem ?

[AdkAdi]

I have tried selecting random numbers and then proving LHS=RHS but I suppose, that is hota legit method!

 

[Steven G]

What are the roots of the given equations?

 

[AdkAdi]

What are the roots of the given equations?

(-1±√(1-4BC))/2 for the first one and for the rest replace BC with AC and AB

 

[Deleted member 4993]

(-1±√(1-4BC))/2 for the first one and for the rest replace BC with AC and AB

Under what conditions, the given quadratic will have

real roots

Imaginary roots?

 

[AdkAdi]

Under what conditions, the given quadratic will have

real roots

Imaginary roots?

If 4BC will be greater than equal to 1 then real roots and if less than 1 imaginary roots

 

[Steven G]

Let's see if I understand what you are saying. Assume 4BC>1. Then 1 - (more than 1) is positive, making the root reals??

 

[AdkAdi]

Let's see if I understand what you are saying. Assume 4BC>1. Then 1 - (more than 1) is positive, making the root reals??

Sorry I messed it up ..I typed the wrong statement..I do know about the nature of roots and the properties of determinants

 

[Steven G]

Sorry I messed it up ..I typed the wrong statement..I do know about the nature of roots and the properties of determinants

Good, so please state the nature of the roots. Hint- sometimes you get exactly one real root (when?), sometimes you get two two real roots (when) and other times you get two imaginary roots (when?).

How do determinants come into this problem?

 

[AdkAdi]

See when the discriminant (b²-4ac=D)
D=0 real equal roots
D>0 Real distinct roots
D<0 Imaginary roots

 

[Steven G]

You really should say how many roots.
Saying D>0 implies Real distinct roots does not say that there are exactly two distinct roots.

 

[Steven G]

Sorry, but I did not think that you meant discriminant when you wrote determinants

 

[AdkAdi]

Sorry, but I did not think that you meant discriminant when you wrote determinants

Yes exactly..?
I do get confused sometimes

 

[AdkAdi]

You really should say how many roots.
Saying D>0 implies Real distinct roots does not say that there are exactly two distinct roots.

If a quadratic is given the. It is implied that there will be 2 roots

 

[Steven G]

If a quadratic is given the. It is implied that there will be 2 roots

Not if there is one root.

 

[AdkAdi]

Not if there is one root.

By the Fundamental Theorem of Algebra if a polynomial has n as it's highest power, then it will have n roots, where a root can be repeated.

 

[topsquark]

By the Fundamental Theorem of Algebra if a polynomial has n as it's highest power, then it will have n roots, where a root can be repeated.

You are absolutely correct but most of us have a tendency to write 1 root if the root is repeated.

-Dan

 

[JeffM]

The first thing I would have done in this problem is NAME THE VARIABLES, and note that they are distinct

[math]0 < p < q < r < s \text { and all are real numbers.}[/math]
If all three have real roots then

[math]1 \ge 4BC,\ 1 \ge 4AC, \text { and } 1 \ge 4AB.[/math]
What then can we say about q and r, given that they are both positive and q < r?