Question

Short-Answer Questions

Solve: $x^2+5x-\left(a^2+a-6\right)=0$ [CBSE 2015]

Answer 1

$$\begin{gathered} x^2+5x-\left(a^2+a-6\right)=0 \\ \Rightarrow x^2+5x-\left(a+3\right)\left(a-2\right)=0 \\ \Rightarrow x^2+\left[\left(a+3\right)-\left(a-2\right)\right]x-\left(a+3\right)\left(a-2\right)=0 \\ \Rightarrow x^2+\left(a+3\right)x-\left(a-2\right)x-\left(a+3\right)\left(a-2\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x\left[x+\left(a+3\right)\right]-\left(a-2\right)\left[x+\left(a+3\right)\right]=0 \\ \Rightarrow \left[x+\left(a+3\right)\right]\left[x-\left(a-2\right)\right]=0 \\ \Rightarrow x+\left(a+3\right)=0\mathrm{or}x-\left(a-2\right)=0 \\ \Rightarrow x=-\left(a+3\right)\mathrm{or}x=\left(a-2\right) \end{gathered}$$

Hence, $-\left(a+3\right)$ and $\left(a-2\right)$ are the roots of the given equation.