Question

Consider the following system of equations:
10 + y = 5x + x2
5x + y = 1
The first equation is an equation of a
✔ parabola
.

The second equation is an equation of a
✔ line
.

How many possible numbers of solutions are there to the system of equations?

0

1

2

3

4

infinite

Answer 1

A line and a (nondegenerate) parabola can intersect in 0, 1, or 2 points only.

Reason: Substituting the line into the parabola gives a quadratic in x, so there are at most 2 real solutions (discriminant <0 → 0, =0 → 1 tangent, >0 → 2).

Examples:
- 0 intersections: y = x^2 and the line y = -1 (no real solution).
- 1 intersection (tangent): y = x^2 and the line y = 0 (double root x = 0).
- 2 intersections: the given system yields x = 1 and x = −11.

3, 4, and infinite are not possible.