Interval in which the Roots Lie
In some problems we
want the roots of the equation ax^{2} + bx + c = 0 to lie in a given
interval. For this we impose conditions on a, b and c. Since a >
0, we can take .
f(x) = x^{2} + .
(i) If both the roots
are positive i.e. they lie in (0, ¥),
then the sum of the roots as well as the product of the roots must be positive.
&⇒
a
+ b = and
ab = with
b^{2}  4ac ≥ 0.
Similarly, if both the
roots are negative i.e. they lie in ( ¥, 0) then the sum of the roots must
be negative and the product of the roots must be positive.
i.e. a
+ b = <
0 and ab = with
b^{2} – 4ac ≥ 0.
Both the roots are of
the same sign if a and c are of same sign. Now if b has the same sign that of
a, both roots are negative or else both roots are positive.
If a and c are of
opposite sign both roots are of opposite sign.
(ii) Both the roots are greater than
a given number k if the following three conditions are satisfied D ≥
0, and
f(k) > 0.
(iii) Both the roots
will be less than a given number k if the following conditions are satisfied:
D ≥ 0, <
k and f(k) > 0.
(iv) Both the roots will lie in the
given interval (k_{1}, k_{2}) if the following conditions are
satisfied: D ≥ 0 k_{1} < and
f(k_{1}) > 0, f(k_{2}) > 0.


(v) Exactly one of the
roots lies in the given interval (k_{1}, k_{2}) if f(k_{1})
. f(k_{2}) < 0.
(vi) A given number k
will lie between the roots if f(k) < 0.
In
particular, the roots of the equation will be of opposite signs if 0 lies
between the roots
&⇒ f(0) < 0. .
Example 1
Let x^{2}
 (m  3)x + m = 0(m ∈ R) be a quadratic
equation. Find the values of m for which the roots are.
Solution: