We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

Each of the Diophantine equations $A^4\pm n{B}^3=C^2$ has an infinite number of integral solutions $(A,B,C)$ for any positive integer $n$. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$, $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A,B,C)$ of the Diophantine equation $a{A}^3+c{B}^3=C^2$ for any co-prime integer pair $(a,c)$.