Question

1. when solving a system of equations that includes one linear equation and one quadratic equation, what are the possibilities for the number of solutions? select all that apply. ☐ no solutions ☐ two solutions ☐ one solution clear all

Explanation:

A system with one linear equation (e.g., \(y = mx + b\)) and one quadratic equation (e.g., \(y = ax^2 + bx + c\)) can be analyzed by substituting the linear expression into the quadratic. This forms a quadratic equation \(ax^2 + (b - m)x + (c - b)=0\) (or similar). The number of real solutions depends on the discriminant of this quadratic. If the discriminant is negative, there are no real solutions (the line and parabola don’t intersect). If the discriminant is zero, there’s one real solution (the line is tangent to the parabola). If the discriminant is positive, there are two real solutions (the line intersects the parabola at two points). So all three options (No solutions, One solution, Two solutions) are possible.

Answer:

No solutions, One solution, Two solutions