Statement-1: If $p x^2+q x+r=0$ is a quadratic equation $(p, q, r \in R)$ such that its roots are $\

Question

Statement-1: If $p x^2+q x+r=0$ is a quadratic equation $(p, q, r \in R)$ such that its roots are $\alpha, \beta$ and $p+q+r<0, p-q+r<0$ and $r>0$ then $[\alpha]+[\beta]=-1$ where [.] denotes greatest integer function. \Statement-2: If for any two real numbers $a$ and $b$, function $f(x)$ is such that $f(a) f(b)<0 \Rightarrow f(x)$ has at least one root lying in $(a, b)$

Professor PC Thomas Classes · Answer

Let $f(x)=px^{2}+qx+r$. $f(0)=r>0, f(1)=p+q+r<0, f(-1)=p-q+r<0$. Hence, one root lies in $(-1, 0)$ and the other in $(0, 1)$. $	herefore [\alpha]=-1$ and $[\beta]=0 \Rightarrow [\alpha]+[\beta]=-1$.