F(x)=3x^2-x+7.       G(x)=x^2+4x+14.     How many times in the graphs of these functions intersect

To find how many times the graphs of the functions \( F(x) = 3x^2 - x + 7 \) and \( G(x) = x^2 + 4x + 14 \) intersect, we need to determine when the two functions are equal. This is done by setting \( F(x) = G(x) \) and solving for \( x \).

### Step 1: Set the functions equal
\[
3x^2 - x + 7 = x^2 + 4x + 14
\]

### Step 2: Move all terms to one side
\[
3x^2 - x + 7 - (x^2 + 4x + 14) = 0
\]

### Step 3: Simplify the equation
\[
3x^2 - x + 7 - x^2 - 4x - 14 = 0
\]
\[
(3x^2 - x^2) + (-x - 4x) + (7 - 14) = 0
\]
\[
2x^2 - 5x - 7 = 0
\]

### Step 4: Solve the quadratic equation
We can solve the quadratic equation \( 2x^2 - 5x - 7 = 0 \) using the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-7)}}{2(2)}
\]
\[
x = \frac{5 \pm \sqrt{25 + 56}}{4}
\]
\[
x = \frac{5 \pm \sqrt{81}}{4}
\]
\[
x = \frac{5 \pm 9}{4}
\]

### Step 5: Find the solutions
There are two possible solutions:
\[
x = \frac{5 + 9}{4} = \frac{14}{4} = 3.5
\]
\[
x = \frac{5 - 9}{4} = \frac{-4}{4} = -1
\]

### Step 6: Conclusion
The graphs of the functions \( F(x) \) and \( G(x) \) intersect at two points: \( x = 3.5 \) and \( x = -1 \).

Therefore, the graphs intersect **two times**.

Would you like further details or explanations? 

Here are 5 related questions to expand on this topic:

1. How do you find the exact points of intersection, including the \( y \)-coordinates?
2. Can you explain how to graph both quadratic functions and locate their intersection points visually?
3. What is the significance of the discriminant in determining the number of intersections?
4. How would the result change if the quadratic equation had no real solutions?
5. Could we solve this problem by completing the square instead of using the quadratic formula?

**Tip**: Always check the discriminant (\( b^2 - 4ac \)) when solving quadratic equations—it tells you how many real solutions exist.

F(x)=3x^2-x+7. G(x)=x^2+4x+14. How many times do the graphs of these functions intersect?

Learn how to solve for the intersection points of two quadratic functions, F(x) = 3x^2 - x + 7 and G(x) = x^2 + 4x + 14, using the quadratic formula. This step-by-step guide is ideal for students in Grades 9-11 and covers important concepts like the quadratic equation, discriminants, and graphing intersections.

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