Question

$a,b,c$ are natural numbers, roots of $a{x}^2+bx+c=0$ lie between $-1$ and $0$, find least values of $a,b,c$ when
(i) roots are distinct
(ii) roots may be equal

Answer 1

$a{x}^2+bx+c=0$ $a,b,cϵN$

(i) $Rootsdistinct$

$D=b^2-4ac>0$ & $-1<\frac{-b}{2a}<0$

$\Rightarrow 0<\frac{b}{2a}<1$

$\Rightarrow b<2a$

$f(-1)>0$

$\Rightarrow a-b+c>0$

$\Rightarrow a+c>b$

$ac<\frac{b^2}{49}$

Using $AM\ge GM$,

$a+c\ge 2\sqrt{ac}$

$\Rightarrow b\ge 2\frac{b}{\sqrt{49}}$

$\Rightarrow a\ge 1$

$a+c>b$

$\Rightarrow c>b-9$

& $b<2a$ {given}

$\Rightarrow c>a$,

$\Rightarrow c>1$

$\Rightarrow c_{min}=2$

Now, $b^2>4ac$

$\Rightarrow b^2>24$

$b>2$

$\Rightarrow b_{min}=3$

(ii) Roots may be equal

$\frac{b^2=4ac}{b<2a}$

$a\ge 1$

$b^2=4ac$

$c>a$

$\left\{\begin{array}{c}{a}_{min}=1\\ b_{min}=2\\ c_{min}=2\end{array}\right\}$