Question

Find the value of $k$ for which the given equations has real and equal roots:
(i) $(k-12)x^2+2(k-12)x+2=0$
(ii) $k^2{x}^2-2(k-1)x+4=0$.

Answer 1

Quadratic equations have real and equal roots if discriminant is zero.

$\Rightarrow D=0$

$\left(i\right)$ $\left(2\right(k-12){)}^2-4(2\left)\right(k-12)=0\Rightarrow (k-12{)}^2-2(k-12)=0$

$(k-12)(k-12-2)=0\Rightarrow (k-12)(k-14)=0$

$k=12,k=14$

$\left(ii\right)$ $(-2(k-1){)}^2-4(4\left)\right(k^2)=0\Rightarrow (k-1{)}^2-4k^2=0$

$3k^2+2k-1=0\Rightarrow (3k-1)(k+1)=0$

$k=\frac{1}{3},k=-1$