Question

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

Answer 1

(a) 1 or 4

$$\begin{gathered} \text{It is given that the roots of the equation (}{x}^2\text{+ 2}(k+2)x+9k=0)\text{are equal}\text{.} \\ \therefore (b^2-4ac)=0 \\ \Rightarrow {\left\{2(k+2)\right\}}^2-4\times 1\times 9k=0 \\ \Rightarrow 4(k^2+4k+4)-36k=0 \\ \Rightarrow 4k^2+16k+16-36k=0 \\ \Rightarrow 4k^2-20k+16=0 \\ \Rightarrow k^2-5k+4=0 \\ \Rightarrow k^2-4k-k+4=0 \\ \Rightarrow k(k-4)-(k-4)=0 \\ \Rightarrow (k-4)(k-1)=0 \\ \Rightarrow k=4\text{or}k=1 \end{gathered}$$