In quadratic equations ax^2+bx+c=0 if a and c are having the same sigh, then the sigh of root will be the option are
a. both roots will be positive
b. both roots will be negative
c. one root will be positive and other negative
d. none of these
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Answers
Answer:
Step-by-step explanation:
ax
2
+bx+c=0, where a
=0.
Condition for real roots of a quadratic equation is b
2
≥4ac. This condition has
to be true. As for the second condition to be true, all coefficient should be
positive.
The proof is easy. If you know a little calculus then you can find that a quadratic
function reaches its extremum when x=
2a
−b
. This result can be derived via
rearranging the terms in the form of a(x+p)
2
+q. Also we know that the
extremum is always halfway between the two roots. So when both of the roots
are negative then the extremum should also be negative.
−
2a
b
<0
or,
a
b
>0
or,
a
2
ab
>0
or, ab>0.
So both a and b should have same sign. Without loss of generality it would be
safe to assume that both a and b is positive (if they were negative then multiply
the quadratic by (-1)). The general from of the roots are
x=
2a
−b±
b
2
−4ac
Now we have two cases to consider.
First one is when b
2
−4ac=0: The roots would become automatically zero as
both a and b is zero.
Second one is when b
2
−4ac>0: We need to the behavior of the root
nearer to zero. If both of the roots are less than zero then so should be the nearer
one. As both a and b are positive,so the root nearer to zero would be
2a
−b±
b
2
−4ac
. If this is less than zero then,
−b+
b
2
−4ac
<0
or, b>
b
2
−4ac
or, b
2
>b
2
−4ac
or, ac>0.
So all three of a,b, c have the same sign. This is the condition on the coefficient.
Answer:
C)One root will be positive and other negative