Question

write the condition for the both roots of f(x)=0 to be greater than a given number k​

Answer 1

$\mathbb\blue{CORRECT\:ANSWER}\huge$

$\green{ax2+bx+c=0 , where a≠0 .}$

Condition for real roots of a quadratic equation is b2≥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=−b2a . 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.

−b2a<0

or, ba>0

or, aba2>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 form of the roots are

x=−b±b2−4ac−−−−−−−√2a .

Now we have two cases to consider.

First one is when b2−4ac=0 : The roots would become automatically zero as both a and b is zero.

Second one is when b2−4ac>0 : We need to see 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 −b+b2−4ac−−−−−−−√2a . If this is less than zero then,

−b+b2−4ac−−−−−−−√<0

or, b>b2−4ac−−−−−−−√

or, b2>b2−4ac

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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