Question

If $x^2+2(k+2)x+9k=0$ has a repeated root, then k = ?
(a) 1 or 4
(b) −1 or 4
(c) 1 or −4
(d) −1 or −4

Answer 1

(a) 1 or 4

$$\begin{gathered} \text{It is given that the equation (}{x}^2+2(k+2)x+9k=0)\text{has repeated roots, i}\text{.e}\text{., roots are equal}\text{.} \\ \therefore D=0 \\ \Rightarrow 4{(k+2)}^2-4\times 1\times 9k=0 \\ \Rightarrow 4k^2+16k+16-36k=0 \\ \Rightarrow 4k^2-20k+16=0 \\ \Rightarrow k^2-5k+4=0 \\ \Rightarrow k^2-4k-k+4=0 \\ \Rightarrow k(k-4)-1(k-4)=0 \\ \Rightarrow (k-1)(k-4)=0 \\ \Rightarrow k=1\text{or 4} \end{gathered}$$