Question

The values of k for which the quadratic equation $k{x}^2+1=kx+3x-11x^2$ has real and equal roots are
(a) −11, −3
(b) 5, 7
(c) 5, −7
(d) none of these

Answer 1

(c) 5, −7

The given equation is $k{x}^2+1=kx+3x-11x^2$ which can be written as.

$$\begin{gathered} k{x}^2+11x^2-kx-3x+1= \\ \Rightarrow \left(k+11\right)x^2-\left(k+3\right)x+1=0 \end{gathered}$$

For equal and real roots, the discriminant of $\left(k+11\right)x^2-\left(k+3\right)x+1=0$.

$$\begin{gathered} \therefore {\left(k+3\right)}^2-4\left(k+11\right)=0 \\ \Rightarrow k^2+2k-35=0 \\ \Rightarrow \left(k-5\right)\left(k+7\right)=0 \\ \Rightarrow k=5,-7 \end{gathered}$$

Hence, the equation has real and equal roots when $k=5,-7.$