Technically speaking, this is not a quadratic equation, as the LHS is a "fractional" polynomial (of leading degree $1$).
With some care, you can indeed turn it to a quadratic problem
$$25t^2+10t+1=0$$ for which the resolution methods are well-known. After resolution for $t$, you still need to discuss the relevance of the solutions, knowing that $t=\sqrt x$.
Technically speaking again, finding the roots of a polynomial is always equivalent to factoring because if you have the roots you have the factors and conversely.
And actually you don't solve a quadratic equation by factoring, but by completing the square, in a way or another. A trigonometric solution is also possible, but... never used.
Your second equation can likewise be turned to a quartic one (fourth degree), the resolution of which is more tedious.
It can be achieved by factoring into two quadratic polynomials, which requires the resolution of a cubic equation (third degree). Interestingly, the latter is solved by turning it to a sextic equation (sixth degree, but in fact bi-cubic) that reduces to a quadratic one.
And for some values of the coefficients (when the cubic has three real roots), you can't avoid a recourse to trigonometry !