Question

Given the equations of a straight line f(x) (in slope-intercept form) and a parabola g(x) (in standard form), describe how to determine the number of intersection points, without finding the coordinates of such points. Do not give an example.

Answer 1

Explanation:

Hello, when you try to find the intersection point(s) you need to solve a system like this one

`\begin{cases} y&= m * x + p }\\ y &= a*x^2 +b*x+c }\end{cases}`

So, you come up with a polynomial equation like.

`ax^2+bx+c=mx+p\\\\ax^2+(b-m)x+c-p=0`

And then, we can estimate the discriminant.

`\Delta=(b-m)^2-4*a*(c-p)`

If
`\Delta<0` there is no real solution, no intersection point.

If
`\Delta=0` there is one intersection point.

If
`\Delta>0` there are two real solutions, so two intersection points.

Hope this helps.