We know that while finding the root of a quadratic equation ax2+bx+c=0 by quadratic formula x=−b±√b2−4ac2a,
if b2−4ac>0, then the roots are real and distinct
if b2−4ac=0, then the roots are real and equal and
if b2−4ac<0, then the roots are imaginary.
Here, the given quadratic equation x2−mx+9=0 is in the form ax2+bx+c=0 where a=1,b=−m and c=9.
(i) If the roots are equal then b2−4ac=0, therefore,
b2−4ac=0⇒(−m)2−(4×1×9)=0⇒m2−36=0⇒m2=36⇒m=±√36⇒m=±6
(ii) If the roots are distinct then b2−4ac>0, therefore,
b2−4ac>0⇒(−m)2−(4×1×9)>0⇒m2−36>0⇒m2>36⇒m>±√36⇒m>±6
(iii) If the roots are imaginary then b2−4ac<0, therefore,
b2−4ac<0⇒(−m)2−(4×1×9)<0⇒m2−36<0⇒m2<36⇒m<±√36⇒m<±6
Hence m=±6 if the roots are equal, m>±6 if the roots are distinct and m<±6 if the roots are imaginary.