Find the values of for which the equation

is true

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- June 17th 2009, 11:07 AMgreat_mathFind the value of k:
Find the values of for which the equation

is true - June 17th 2009, 11:56 AMrunning-gag
- June 18th 2009, 03:52 AMAmer
- June 18th 2009, 07:13 AMAmer
- June 18th 2009, 07:20 AMSoroban
Hello, great_math!

A puzzling problem . . . What is meant by "is true"?

Quote:

Find the values of for which the equation: . .is true.

. .

. .

Quadratic Formula: .

Now it all depends on what they're asking for . . .

. . The discriminant must be nonnegative: .

. . Since

*Good luck!*

- June 19th 2009, 06:22 PMpankaj
Putting .What we have is the quadratic

where

Discriminant =

Thus we will get real values of for all real values of .

But we are required to ensure that these values must lie on the interval

Let

__CASE 1__

If exactly one root of lies on (-1,1),then we must have

Now,

,thus

or

or i.e.

__CASE 2__

If both roots on the interval then following three conditions must be satisfied:

(1)

(2)

(3) ; and being the rootsof the equation

implies that .

Thus .

implies that

(Recall that if and are roots of then )

On solving for we get

Taking intersection of solutions of (1),(2) and (3),we get

Finally taking union ofand__Case 1__and also checking for , we arrive at the conclusion that the given equation is true for all real .__Case 2__

must be the answer.