Location of Roots

Case 1. For a number k.             Both roots are less than k.
1.D>=0 ( roots may be equal )
2. af(k) > 0
3. K > -b/2a  where a < β

 
Case 2. For a number k.
            Both roots are greater  than k.
Case 2. For a number k.             Both roots are greater  than k.
1.D>=0 ( roots may be equal )
2. af(k) > 0
3. K < -b/2a  where a < β

Case 3. For a number k.
           k lies between the roots.
Case 3. For a number k.            k lies between the roots.

1.D > 0

2. af(k) < 0

where a < β


Case 4. For a number k1 & k2.

           exactly one root lies in the interval ( k1 , k2).

Case 4. For a number k1 & k2.            exactly one root lies in the interval ( k1 , k2).

1.D > 0 ( roots may be equal )

2. f(k1) f(k1) < 0

where a < β


Case 5. For a number k1 & k2.
           Both roots are lies between  k1 & k2.
Case 5. For a number k1 & k2.            Both roots are lies between  k1 & k2.

1.D>=0 ( roots may be equal )

2. af(k1) > 0 & af(k2) > 0

3. k1 < -b/2a  < k2     where a <= β  &  k1 < k2.


Case 6. For a number k1 & k2.

           k1 & k2 are lies between  roots.

Case 6. For a number k1 & k2.            k1 & k2 are lies between  roots.

1.D > 0

2. af(k1) < 0 & af(k2) < 0

    where a <  β