Math Problem Statement
F(x)=3x^2-x+7. G(x)=x^2+4x+14. How many times in the graphs of these functions intersect
Solution
To find how many times the graphs of the functions and intersect, we need to determine when the two functions are equal. This is done by setting and solving for .
Step 1: Set the functions equal
Step 2: Move all terms to one side
Step 3: Simplify the equation
Step 4: Solve the quadratic equation
We can solve the quadratic equation using the quadratic formula:
Step 5: Find the solutions
There are two possible solutions:
Step 6: Conclusion
The graphs of the functions and intersect at two points: and .
Therefore, the graphs intersect two times.
Would you like further details or explanations?
Here are 5 related questions to expand on this topic:
- How do you find the exact points of intersection, including the -coordinates?
- Can you explain how to graph both quadratic functions and locate their intersection points visually?
- What is the significance of the discriminant in determining the number of intersections?
- How would the result change if the quadratic equation had no real solutions?
- Could we solve this problem by completing the square instead of using the quadratic formula?
Tip: Always check the discriminant () when solving quadratic equations—it tells you how many real solutions exist.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Graph Intersections
Formulas
Quadratic equation: ax^2 + bx + c = 0
Quadratic formula: x = [-b ± √(b^2 - 4ac)] / 2a
Theorems
Quadratic Formula
Discriminant of a Quadratic Equation
Suitable Grade Level
Grades 9-11
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