if both roots of the quadratic ax^2+bx+c=0are negative then discuss the sign of a,b,c.
Answers
Answer:
Step-by-step explanation:
A will be positive
B will be positive
C will be positive
Answer:
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.