## if both roots of the quadratic ax^2+bx+c=0are negative then discuss the sign of a,b,c.​

Question

if both roots of the quadratic ax^2+bx+c=0are negative then discuss the sign of a,b,c.​

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4 weeks 2021-08-16T05:23:20+00:00 2 Answers 0 views 0

Step-by-step explanation:

A will be positive

B will be positive

C will be positive

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.