Math, asked by avinashjk07, 3 months ago

Matrix Matching questions :
If f(x) = ax? + bx + c = 0
Column - 1
Column - II
3.
If both roots of f(x) = 0 are positive
a)->
4.
If both roots of f(x) = 0 are negative then
b) > 0
5.
If roots of f(x) = 0 are opposite signs then
c) <0
b
6.
If roots of f(x) = 0 are numerically equal but
d) -<0
opposite signs then
EMERALD PACKAGE
One or more than one correct answer tyne questionsIf f(x) = ax? + bx + c = 0
Column - 1
Column - II
3.
If both roots of f(x) = 0 are positive
a)->
4.
If both roots of f(x) = 0 are negative then
b) > 0
5.
If roots of f(x) = 0 are opposite signs then
c) <0
b
6.
If roots of f(x) = 0 are numerically equal but
d) -<0
op​

Answers

Answered by aslamm1
1

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.  

Step-by-step explanation:

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