Interval in which the Roots Lie
In some problems we
want the roots of the equation ax2 + 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) = x2 + 
.
(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
b2 - 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
b2 – 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 (k1, k2) if the following conditions are
satisfied: D ≥ 0 k1 < -
and
f(k1) > 0, f(k2) > 0.
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(v) Exactly one of the
roots lies in the given interval (k1, k2) if f(k1)
. f(k2) < 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 x2
- (m - 3)x + m = 0(m ∈ R) be a quadratic
equation. Find the values of m for which the roots are.
Solution:
