Question

find the discriminant for the quadratic equation 2(p2+q2) x2 +2(p+q)x + 1 =0. (p is not equal to q).. Also determine the nature of the roots.


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Answer 1

$\textbf{Given:}$

$2(p^2+q^2)x^2+2(p+q)x+1=0$

$\textbf{To find:}$

$\text{Discriminant of the given equation and}$

$\text{nature of roots}$

$\textbf{Solution:}$

$\text{Consider,}$

$2(p^2+q^2)x^2+2(p+q)x+1=0$

$\text{Here, a=2(p^2+q^2) , b=2(p+q) , c=1 }$

$\textbf{Discriminant}$

$=b^2-4ac$

$=[2(p+q)]^2-4(2(p^2+q^2))(1)$

$=4(p+q)^2-8(p^2+q^2)$

$=4(p^2+q^2+2pq)-8(p^2+q^2)$

$=4p^2+4q^2+8pq-8p^2-8q^2$

$=8pq-4p^2-4q^2$

$=-4(p^2+q^2-2pq)$

$=-4(p-q)^2\,<\,0$ $(\because\;p{\neq}q)$

$\therefore\textbf{The roots are not real}$