mark is solving an equation where one side is a quadratic expression and the other side is a linear…

Explanation:

Step1: Recall function types

A quadratic function is of the form $y = ax^{2}+bx + c$ ($a\neq0$) and has a parabolic graph. A linear function is of the form $y=mx + n$ and has a straight - line graph.

Step2: Consider intersection cases

When we find the intersection of $y = ax^{2}+bx + c$ and $y=mx + n$, we set $ax^{2}+bx + c=mx + n$, which simplifies to $ax^{2}+(b - m)x+(c - n)=0$. This is a quadratic equation of the form $Ax^{2}+Bx + C = 0$ (where $A = a$, $B=b - m$, $C=c - n$).

Step3: Use the discriminant

The number of solutions of a quadratic equation $Ax^{2}+Bx + C = 0$ is given by the discriminant $\Delta=B^{2}-4AC$. A non - zero quadratic equation can have at most 2 real solutions. So, the graphs of a quadratic function and a linear function can intersect at most 2 times.

Answer:

two