Question
Find $m$ if $(m-12) x^2+2(m-12) x+2=0$ has real and equal roots.
✓
Answer
$(m-12) x^2-(2 m-24) x+2=0 \text { compare with } a x^2+b x+c=0 $
$\Rightarrow a=m-12, b=-2 m+24 \text { and } c=2 $
$\therefore b^2-4 a c=(-2 m+24)^2-4(m-12)(2) $
$=4 m^2-96 m+576-8 m+96 $
$=4 m^2-104 m+672 $
$=m^2-26 m+168$
If roots are equal and real then, $\therefore b^2-4 a c=0$
$m ^2-26 m+168=0 $
$\Rightarrow m ^2-12 m-14 m+168=0 $
$\Rightarrow m ( m -12)-14(m-12)=0 $
$\Rightarrow( m -12)( m -14)=0 $
$m=12 \text { or } m =14$
$m = 12 \ or\ m = 14$
$\Rightarrow a=m-12, b=-2 m+24 \text { and } c=2 $
$\therefore b^2-4 a c=(-2 m+24)^2-4(m-12)(2) $
$=4 m^2-96 m+576-8 m+96 $
$=4 m^2-104 m+672 $
$=m^2-26 m+168$
If roots are equal and real then, $\therefore b^2-4 a c=0$
$m ^2-26 m+168=0 $
$\Rightarrow m ^2-12 m-14 m+168=0 $
$\Rightarrow m ( m -12)-14(m-12)=0 $
$\Rightarrow( m -12)( m -14)=0 $
$m=12 \text { or } m =14$
$m = 12 \ or\ m = 14$
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