Question

If roots of $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$ and $4a+2b+c>0,4a-2b+c>0$, and $c<0$, then possible value/values of $\left[\alpha \right]+\left[\beta \right]$ is/are (where $[\cdot ]$ represents greatest integer function)

• A. $-2$
• B. $-1$
• C. $0$
• D. $1$

Answer 1

Answer: (A), (B), (C)

$-1$
$0$
$-2$

Given,$a{x}^2+bx+c=0$$4a+2b+c>0\Rightarrow f\left(2\right)>0$$4a-2b+c>0\Rightarrow f(-2)>0$$c<0\Rightarrow f\left(0\right)<0$$\Rightarrow f\left(0\right).f\left(2\right)<0$ $\Rightarrow$ one root say $\alpha$ lies between $0$ and $1$
$f\left(0\right).f(-2)<0$ $\Rightarrow$ one root say $\beta$ lies between $0<\alpha <2$$\Rightarrow 0⩽\left[\alpha \right]⩽1$$-2<\beta <0$$\Rightarrow -2⩽\left[\beta \right]⩽-1$$\Rightarrow 0-2⩽\left[\alpha \right]+\left[\beta \right]⩽1-1$$\Rightarrow -2⩽\left[\alpha \right]+\left[\beta \right]⩽0$$\therefore$ Possible values are $-2,-1,0$Hence options A, B, C are correct.