Question:
Find the value of $k$ for which the equation $x^{2}+k(2 x+k-1)+2=0$ has real and equal roots.
Solution:
Let $x^{2}+k(2 x+k-1)+2=0$ be a quadratic equation.
$x^{2}+k(2 x+k-1)+2=0$
$x^{2}+2 x k+k^{2}-k+2=0$
It is given that, it has real and equal roots.
$\Rightarrow$ Discriminant $=0$
$\Rightarrow b^{2}-4 a c=0$
$\Rightarrow(2 k)^{2}-4(1)\left(k^{2}-k+2\right)=0$
$\Rightarrow 4 k^{2}-4 k^{2}+4 k-8=0$
$\Rightarrow 4 k=8$
$\Rightarrow k=2$
Hence, the value of k is 2.
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