Question
If $$\ell ,m,n$$ are real, $$\ell \ne m,$$ then the roots by the equation: $$\left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0$$ are
A.
Real and equal
B.
Complex
C.
Real and unequal
D.
None of these
Answer :
Real and unequal
Solution :
$$\eqalign{
& \ell ,m,n\,\,{\text{are real, }}\ell \ne m \cr
& {\text{Given equation is }} \cr
& \left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0 \cr
& D = 25{\left( {\ell + m} \right)^2} + 8{\left( {\ell - m} \right)^2} > 0,\ell ,m \in R \cr
& \therefore \,\,\,{\text{Roots are real and unequal}}{\text{.}} \cr} $$