Question

a b c are natural
numbers, and roots of
ax^2+ bx+c= 0 lie between-1 and 0, then the
square root of the product of least values of a, b, c is​

Answer 1

Answer:ax  

2

+bx+c=0           a,b,cϵN

(i)  Roots distinct

D=b  

2

−4ac>0 & −1<  

2a

−b

​  

<0

⇒0<  

2a

b

​  

<1

⇒b<2a

f(−1)>0

⇒a−b+c>0

⇒a+c>b

ac<  

49

b  

2

 

​  

 

Using AM≥GM,

a+c≥2  

ac

​  

 

⇒b≥2  

49

​  

 

b

​  

 

⇒a≥1

a+c>b

⇒c>b−9

& b<2a {given}

⇒c>a,

⇒c>1

⇒c  

min

​  

=2

Now, b  

2

>4ac

⇒b  

2

>24

b>2

⇒b  

min

​  

=3

(ii)  Roots may be equal

b<2a

b  

2

=4ac

​  

 

a≥1

b  

2

=4ac

c>a

⎩

⎪

⎪

⎨

⎪

⎪

⎧

​  

 

a  

min

​  

=1

b  

min

​  

=2

c  

min

​  

=2

​  

 

⎭

⎪

⎪

⎬

⎪

⎪

⎫

​

Step-by-step explanation: