Question:
The values of $k$ for which the quadratic equation $k x^{2}+1=k x+3 x-11 x^{2}$ has real and equal roots are
(a) −11, −3
(b) 5, 7
(c) 5, −7
(d) none of these
Solution:
(c) 5, −7
The given equation is $k x^{2}+1=k x+3 x-11 x^{2}$ which can be written as.
$k x^{2}+11 x^{2}-k x-3 x+1=$
$\Rightarrow(k+11) x^{2}-(k+3) x+1=0$
For equal and real roots, the discriminant of $(k+11) x^{2}-(k+3) x+1=0$.
$\therefore(k+3)^{2}-4(k+11)=0$
$\Rightarrow k^{2}+2 k-35=0$
$\Rightarrow(k-5)(k+7)=0$
$\Rightarrow k=5,-7$
Hence, the equation has real and equal roots when $k=5,-7$.