Question

Can the quadratic polynomial $x^3+kx+k$ have equal zeroes for some odd integer $k>1$ ?

Answer 1

For a quadratic equation to have equal roots ,it must satisfy for condition.

$b^2-4ac=0$

Given equation is $x^2+kx+k=0$

$a=1,b=k,x=k$

So substituting in the equation we get,

$\begin{array}{c}\Rightarrow k^2-4\left(1\right)\left(k\right)=0\\ \Rightarrow k^2-4k=0\\ \Rightarrow k\left(k-4\right)=0\\ \therefore k=0,k=4\end{array}$

It is given that $k$ is greater than $1$.