text: # Find the value of α for which the equation
  
  Question:
  
  Find the value of a for which the equation $(\alpha-12) x^{2}+2(\alpha-12) x+2=0$ has equal roots.
  
  Solution:
  
  Given:
  
  $(\alpha-12) x^{2}+2(\alpha-12) x+2=0$
  
  Here,
  
  $a=(\alpha-12), b=2(\alpha-12)$ and $c=2$
  
  It is given that the roots of the equation are equal; therefore, we have:
  
  $D=0$
  
  $\Rightarrow\left(b^{2}-4 a c\right)=0$
  
  $\Rightarrow\{2(\alpha-12)\}^{2}-4 \times(\alpha-12) \times 2=0$
  
  $\Rightarrow 4\left(\alpha^{2}-24 \alpha+144\right)-8(\alpha-12)=0$
  
  $\Rightarrow 4 \alpha^{2}-96 \alpha+576-8 \alpha+96=0$
  
  $\Rightarrow 4 \alpha^{2}-104 \alpha+672=0$
  
  $\Rightarrow \alpha^{2}-26 \alpha+168=0$
  
  $\Rightarrow \alpha^{2}-14 \alpha-12 \alpha+168=0$
  
  $\Rightarrow \alpha(\alpha-14)-12(\alpha-14)=0$
  
  $\Rightarrow(\alpha-14)(\alpha-12)=0$
  
  $\therefore \alpha=14$ or $\alpha=12$
  
  If the value of $\alpha$ is 12, the given equation becomes non-quadratic.
  
  Therefore, the valueof $\alpha$ will be 14 for the equation to have equal roots.