1. The roots of are find the values of .
That the roots are means that:
expanding the brackets gives:
so
RonL
Solve
First:
There are critical points where the numerator is 0 and where the denominator is 0. So we have critical points at:
x + 3 = 0 --> x = -3
x - 4 = 0 --> x = 4
So break the real line up into and test each interval:
: (Check!)
: (Nope!)
: (Check!)
So the solution set is
-Dan
(Hi CaptainBlack! )
For determine all values of such that .
Now for large as the leading term is dominant. So the question is realy asking us to find the values of k such that has no real roots.
From the quadratic formula we know the roots of are:
,
and these are not real if , which is equivalent to and .
RonL
Hello, SportfreundeKeaneKent!
If the roots of the equation are , and ,
find the values of , and
If , and are roots of the cubic,
. . then are factors of the cubic.
Hence, the cubic is: .
Therefore: .
Solve: . . where
Multiply both sides by .
If or ,
. . we have: .
The "stronger" inequality is: .
If or ,
. . we have: .
The "stronger" inequality is: .
Solution: .
For , determine all the values of such that
If , the parabola is completely above the x-axis.
If it had x-intercepts, they would occur where .
. . That is: .
Quadratic Formula: .
To have no x-intercepts, the discriminant must be negative.
. . That is: .
Complete the square: .
And we have: .
Hence: .