Answer :
Given the system of equations where \( f(x) \) is a quadratic function and \( g(x) \) is a linear function:
[tex]\[ \begin{array}{l} y = f(x) \\ y = g(x) \end{array} \][/tex]
We need to determine the number of possible solutions \( (x, y) \) that satisfy both equations simultaneously. Let's analyze the intersection points of these functions.
1. Intersection Points:
- A quadratic function \( f(x) \) is expressed as \( f(x) = ax^2 + bx + c \), where \( a \neq 0 \).
- A linear function \( g(x) \) is expressed as \( g(x) = mx + n \).
2. Finding Solutions:
- To find the intersection points, set \( f(x) \) equal to \( g(x) \):
[tex]\[ ax^2 + bx + c = mx + n. \][/tex]
Simplifying, we get:
[tex]\[ ax^2 + (b - m)x + (c - n) = 0. \][/tex]
This is a quadratic equation in \( x \).
3. Nature of Solutions:
- The number of solutions to this quadratic equation depends on its discriminant \( \Delta \):
[tex]\[ \Delta = (b - m)^2 - 4a(c - n). \][/tex]
- The discriminant can tell us the number of real solutions:
- If \( \Delta > 0 \): There are 2 distinct real solutions (two intersection points).
- If \( \Delta = 0 \): There is exactly 1 real solution (one intersection point, also called a tangent point).
- If \( \Delta < 0 \): There are no real solutions (no intersection points).
4. Conclusion:
- Based on the analysis of the discriminant, the quadratic function and linear function can intersect at:
- No points (when \( \Delta < 0 \)),
- Exactly one point (when \( \Delta = 0 \)),
- Exactly two points (when \( \Delta > 0 \)).
Thus, the correct statement that describes the number of possible solutions to the system is:
A. The system may have no, 1, or 2 solutions.
[tex]\[ \begin{array}{l} y = f(x) \\ y = g(x) \end{array} \][/tex]
We need to determine the number of possible solutions \( (x, y) \) that satisfy both equations simultaneously. Let's analyze the intersection points of these functions.
1. Intersection Points:
- A quadratic function \( f(x) \) is expressed as \( f(x) = ax^2 + bx + c \), where \( a \neq 0 \).
- A linear function \( g(x) \) is expressed as \( g(x) = mx + n \).
2. Finding Solutions:
- To find the intersection points, set \( f(x) \) equal to \( g(x) \):
[tex]\[ ax^2 + bx + c = mx + n. \][/tex]
Simplifying, we get:
[tex]\[ ax^2 + (b - m)x + (c - n) = 0. \][/tex]
This is a quadratic equation in \( x \).
3. Nature of Solutions:
- The number of solutions to this quadratic equation depends on its discriminant \( \Delta \):
[tex]\[ \Delta = (b - m)^2 - 4a(c - n). \][/tex]
- The discriminant can tell us the number of real solutions:
- If \( \Delta > 0 \): There are 2 distinct real solutions (two intersection points).
- If \( \Delta = 0 \): There is exactly 1 real solution (one intersection point, also called a tangent point).
- If \( \Delta < 0 \): There are no real solutions (no intersection points).
4. Conclusion:
- Based on the analysis of the discriminant, the quadratic function and linear function can intersect at:
- No points (when \( \Delta < 0 \)),
- Exactly one point (when \( \Delta = 0 \)),
- Exactly two points (when \( \Delta > 0 \)).
Thus, the correct statement that describes the number of possible solutions to the system is:
A. The system may have no, 1, or 2 solutions.
