Inconsistent solution for the factorized equation When I try to solve this equation as follows:
\begin{aligned}-k_1^2 \, \cos^2\theta \, - 2 \left( k_1k_5 + k_4 \right) \cos\theta+k_4^2-k_5^2+1 \geqslant 0   \end{aligned}
Noting that $k_1$, $k_4$, $k_5$ are real constants. I assumed:
$$
\left\{ \matrix{
  x=\cos\theta \hfill \cr 
  a=-k_1^2 \hfill \cr 
  b=-2 \left( k_1k_5 + k_4 \right)  \hfill \cr
c=k_4^2-k_5^2+1 \hfill \cr }  \right.
$$
then it will be:
\begin{aligned}a \, x^2 +b\,x +c  \geqslant 0   \end{aligned}
then finding the roots as follows:
\begin{aligned}x_1,_2 =\frac{-b\pm\sqrt{b^2-4ac}}{2a}   \end{aligned}
led to:
\begin{aligned}\left(\cos\theta +\frac{k_1k_5 + k_4+\sqrt{k_1^2+k_4^2+k_1^2k_4^2+2k_1k_4k_5              }}{k_1^2}  \right)\\\left(\cos\theta +\frac{k_1k_5 + k_4-\sqrt{k_1^2+k_4^2+k_1^2k_4^2+2k_1k_4k_5              }}{k_1^2}  \right) \geqslant 0\end{aligned}
The problem I'm facing that when substituting real values, I cannot have the same result out of the new equation, for example susbstituting:
$$
\left\{ \matrix{
  \theta=0.17453292 \hfill \cr 
  k_1=3.5 \hfill \cr 
  k_4=1.75\hfill \cr
k_5=-3.75 \hfill \cr }  \right.
$$
I get in the main equation this:
\begin{aligned}0.52375908\geqslant 0\end{aligned}
and in the factored equation, I get this:
\begin{aligned}-0.04275584\geqslant 0\end{aligned}
Please advise me – what is wrong in my solution?

[A Quadratic equation][1]
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, one says that it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation
\begin{aligned}a \, x^2 +b\,x +c  =  a\left( x-x_1\right)\left( x-x_2\right)  \end{aligned}
So, the factored equation should have been:
\begin{aligned}-k_1^2\left(\cos\theta +\frac{k_1k_5 + k_4+\sqrt{k_1^2+k_4^2+k_1^2k_4^2+2k_1k_4k_5              }}{k_1^2}  \right)\\\left(\cos\theta +\frac{k_1k_5 + k_4-\sqrt{k_1^2+k_4^2+k_1^2k_4^2+2k_1k_4k_5              }}{k_1^2}  \right) \geqslant 0\end{aligned}
[1]: https://en.wikipedia.org/wiki/Quadratic_equation
