Question
Solve the following quadratic equation:
$ x^2+5 x-\left(a^2+a-6\right)=0 $
$ x^2+5 x-\left(a^2+a-6\right)=0 $
✓
Answer
$ x^2+5 x-\left(a^2+a-6\right)=0 $
$ \Rightarrow x^2+5 x-\left(a^2+3 a-2 a-6\right)=0 $
$ \Rightarrow x^2+5 x-[a(a+3)-2(a+3)]=0 $
$ \Rightarrow x^2+5 x-[(a+3)(a-2)]=0 $
$ \Rightarrow x^2+(a+3) x-(a-2) x-(a+3)(a-2)=0 $
$ \Rightarrow x[x+(a+3)]-(a+2)[x+(a+3)]=0 $
$ \Rightarrow[x+(a+3)][x-(a-2)]=0 $
$ \Rightarrow x+(a+3)=0 \text { or } x-(a-2)=0 $
$ \Rightarrow x=-(a+3) \text { or } x=a-2$
$ \Rightarrow x^2+5 x-\left(a^2+3 a-2 a-6\right)=0 $
$ \Rightarrow x^2+5 x-[a(a+3)-2(a+3)]=0 $
$ \Rightarrow x^2+5 x-[(a+3)(a-2)]=0 $
$ \Rightarrow x^2+(a+3) x-(a-2) x-(a+3)(a-2)=0 $
$ \Rightarrow x[x+(a+3)]-(a+2)[x+(a+3)]=0 $
$ \Rightarrow[x+(a+3)][x-(a-2)]=0 $
$ \Rightarrow x+(a+3)=0 \text { or } x-(a-2)=0 $
$ \Rightarrow x=-(a+3) \text { or } x=a-2$
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