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

Answers ( )

    0
    2021-08-16T05:24:28+00:00

    Answer:

    Step-by-step explanation:

    A will be positive

    B will be positive

    C will be positive

    0
    2021-08-16T05:24:56+00:00

    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.

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