Question

Let $k$ be the value of $\lambda$ for which the given equation will have equal roots : $x^2+2(3\lambda +5)x+2(9{\lambda }^2+25)=0$.Find $6k$ ?

Answer 1

Discriminant of given quadratic is,
$D=4(3\lambda +5{)}^2-8(9{\lambda }^2+25)=-4{(3\lambda -5)}^2.$
If $\lambda \ne \frac{5}{3}$ their D is always-ive and
hence roots are complex.
But if $\lambda =\frac{5}{3}$
then $D=0$ and in that case the root will be equal.