Find the values of for which the equation
is true
Hello, great_math!
A puzzling problem . . . What is meant by "is true"?
Replace with :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!
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 of Case 1 and Case 2 and also checking for , we arrive at the conclusion that the given equation is true for all real .
must be the answer.