Question

If $a{x}^2+bx+c=0,a\ne 0,a,b,c\in R$ has distinct real roots in $(1,2)$ then $a$ and $5a+2b+c$ have

• A. same sign
• B. opposite sign
• C. not detremined
• D. none of these

Answer 1

Answer: (A)

same sign

Let $x_1$ and $x_2$ be two roots of $a{x}^2+bx+c=0$

$1<x_1<2$ and $1<x_2<2$

Now $a(5a+2b+c)=a^2(5+2\frac{b}{a}+\frac{c}{a})$

$=a^2(5+2(-1)(x_1+x_2)+x_1{x}_2)=a^2((x_1-2)(x_2-2)+1)>0$

Hence $a$ and $5a+2b+c$ are of same sign.