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
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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:
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