Question

pablo graphs a system of equations. one equation is quadratic and the other equation is linear. what is the greatest number of possible solutions to this system?
○ 0
○ 1
○ 2
○ 4

Explanation:

Step1: Recall Graph Shapes

A quadratic equation graphs as a parabola (a U - shaped or inverted U - shaped curve), and a linear equation graphs as a straight line.

Step2: Analyze Intersections

To find the number of solutions to a system of equations, we look at the number of intersection points between their graphs. A straight line can intersect a parabola at most 2 times. For example, if we have the quadratic equation \(y = x^{2}\) (a parabola opening upwards) and the linear equation \(y=x + 2\), we can solve \(x^{2}=x + 2\), which simplifies to \(x^{2}-x - 2=0\). Factoring gives \((x - 2)(x+1)=0\), so \(x = 2\) or \(x=-1\), which means two intersection points. A line can also intersect a parabola at 0 (if the line is far from the parabola) or 1 (if the line is tangent to the parabola) points, but the maximum number of intersections is 2.

Answer:

2 (the option corresponding to 2, e.g., if the option with 2 is labeled as C, then C. 2)