Thank you for reporting, we will resolve it shortly
Q.
The value of $k$ so that the equations $x^{2}-x-12=0$ and $k x^{2}+10 x+3=0$ may have one root in common, is
Complex Numbers and Quadratic Equations
Solution:
Let $\alpha$ be the common root
Then, $\alpha^{2}-\alpha-12=0 $
and $k \alpha^{2}+10 \alpha+3=0$
Solving the two equations, we get
$\frac{\alpha^{2}}{117}=\frac{\alpha}{-12 k-3}=\frac{1}{10+k}$
$\Rightarrow (-12 k-3)^{2}=117(10+k)$
$\Rightarrow 9(4 k+1)^{2}=117(10+k)$
$\Rightarrow (4 k+1)^{2}=13(10+k)$
$\Rightarrow 16 k^{2}+8 k+1=130+13 k$
$\Rightarrow 16 k^{2}-5 k-129=0$
$\Rightarrow 16 k^{2}-48 k+43 k-129=0$
$\therefore k=3 $
or $k=\frac{-43}{16}$