ISI2015-DCG-30

Question

Let $p,q,r,s$ be real numbers such that $pr=2(q+s)$. Consider the equations $x^2+px+q=0$ and $x^2+rx+s=0$. Then

• A. at least one of the equations has real roots
• B. both these equations have real roots
• C. neither of these equations have real roots
• D. given data is not sufficient to arrive at any conclusion

Answer 1

The discriminant of $x^2+px+q=0$ is $Δ_1=p^2−4q$

and that of $x^2+rx+s=0$ is $Δ_2=r^2−4s$

and $Δ_1+Δ_2=p^2−4q+r^2−4s$

$= p^2+r^2−4(q+s)$

$= (p+r)^2−2pr−4(q+s)$

$= (p+r)^2−2[pr−2(q+s)]$

and if $pr=2(q+s)$, we have $Δ_1+Δ_2=(p+r)^2$

Since, the sum of two discriminants is positive, so at least one of them has to be positive.

Therefore, at least one the equation $x^2+px+q  = 0 $ and $x^2+rx + s = 0 $ has the real roots.