Question

The quadratic equation $\frac{11}{4}{x}^2-11(p+q)x+(10p^2+24pq+10q^2)=0,$ where $p\ne \pm q$ has

• A. Real and equal roots
• B. Real and distinct roots
• C. No real roots
• D. One real and one non-real root

Answer 1

The correct option is B Real and distinct roots

The given quadratic equation is $\frac{11}{4}{x}^2-11(p+q)x+(10p^2+24pq+10q^2)=0,p\ne \pm q,$
Comparing this equation with standard quadratic equation $a{x}^2+bx+c=0,$ we get
$a=\frac{11}{4},b=-11(p+q),c=10p^2+24pq+10q^2$
Now, $D=b^2-4ac$
$=[-11(p+q{)}^2-4\times \frac{11}{4}\times \frac{11}{4}\times (10p^2+24pq+10q^3)$
$=11\times 11\times (p+q{)}^2-11(10p^2+24pq+10q^2)$
$=11(11p^2+22pq+12q^2-10p^2-24pq-10q^2)$
$=11(p^2-2pq+q^2)$
$=11(p-q{)}^2>0\because p\ne \pm q$
Therefore, roots are real and distinct.
Hence, the correct answer is option (2).