Math Problem Statement
Solution
The image presents a mathematical problem involving the intersection of a straight line and a parabola. The given equations are:
- Line:
- Parabola:
The question asks how many intersection points these two equations have, and provides multiple-choice options:
A. Two distinct intersection points
B. Four distinct intersection points
C. No intersection point
D. A unique intersection point
E. Three distinct intersection points
Solution:
-
Rearrange the line equation:
From , solve for : -
Substitute this into the parabola equation:
Set , so: Multiply both sides by 2 to eliminate the fraction: Simplify: Expand: Rearrange to form a quadratic equation: -
Solve the quadratic equation:
We now solve . The discriminant of this quadratic equation is: Since the discriminant is negative, there are no real solutions.
Thus, the line and the parabola do not intersect at any point.
Conclusion:
The correct answer is C. No intersection point.
Would you like further details or have any other questions?
Here are 5 related questions to explore:
- How do you solve systems involving non-linear equations geometrically?
- What does a negative discriminant mean in terms of intersection points?
- How can you find the vertex of a parabola from its equation?
- How do different types of conic sections intersect with lines?
- What methods can be used to solve quadratic equations?
Tip: Always check the discriminant when solving quadratic equations to determine if real solutions exist
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Intersection of Curves
Formulas
Quadratic equation ax^2 + bx + c = 0
Discriminant formula Δ = b^2 - 4ac
Theorems
Quadratic formula
Discriminant Analysis for Intersection
Suitable Grade Level
Grades 10-12
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