Question

If $p,q,r\in R$ and the quadratic equation $p{x}^2+qx+r=0$ has no real root, then

• A. $p(p+q+r)>0$
• B. $r\left(p+q+r\right)>0$
• C. $q(p+q+r)>0$
• D. $(p+q+r)>0$

Answer 1

The correct option is B

$r\left(p+q+r\right)>0$

Explanation for the correct option

Step 1: Solve the above quadratic equation for no real root

Given information

$\Rightarrow p{x}^2+qx+r=0$ $p,q,r\in R$

Let $f\left(x\right)=p{x}^2+qx+r=0$

If the quadratic equation has no real root then,

$\therefore f\left(x\right)=0$

Step 2: And $f\left(x\right)$ will have the same sign as that of $p$ for all $x\in R$

Put $x=1$

$\begin{array}{rcl}\therefore p.f\left(1\right)& >& 0=p(p+q+r)>0\end{array}$

Now for $x=0and1$

$\Rightarrow f\left(0\right).f\left(1\right)>0=r(p+q+r)>0$

Hence option (B) is the correct answer.