F(x)=3x^2-x+7.       G(x)=x^2=4x+14.     How many times will 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 the values of \( x \) for which \( F(x) = G(x) \). This means we set the two functions equal to each other:

\[
3x^2 - x + 7 = x^2 - 4x + 14
\]

Now, we simplify this equation by moving all terms to one side:

\[
3x^2 - x + 7 - x^2 + 4x - 14 = 0
\]

Combine like terms:

\[
2x^2 + 3x - 7 = 0
\]

This is a quadratic equation. To determine how many times the graphs intersect, we need to find the number of real solutions to this quadratic equation. We can use the discriminant formula, which is given by:

\[
\Delta = b^2 - 4ac
\]

where \( a = 2 \), \( b = 3 \), and \( c = -7 \). Plugging in these values:

\[
\Delta = (3)^2 - 4 \cdot 2 \cdot (-7)
\]

\[
\Delta = 9 + 56
\]

\[
\Delta = 65
\]

Since the discriminant \(\Delta = 65\) is positive, there are 2 real solutions to the quadratic equation.

### Conclusion:
The graphs of the functions \( F(x) = 3x^2 - x + 7 \) and \( G(x) = x^2 - 4x + 14 \) will intersect **2 times**.

Would you like to see the specific points where these graphs intersect, or do you have any other questions?

### Related Questions:

1. What are the exact points of intersection for the functions \( F(x) \) and \( G(x) \)?
2. How do you find the roots of a quadratic equation using the quadratic formula?
3. What is the significance of the discriminant in determining the nature of the roots?
4. How can you graphically represent the intersections of two quadratic functions?
5. What are some real-world applications of finding intersections between functions?

### Tip:
For any quadratic equation \( ax^2 + bx + c = 0 \), remember that the discriminant \( b^2 - 4ac \) tells you the number and type of solutions: if it's positive, there are two distinct real roots; if it's zero, there's one real root; and if it's negative, there are no real roots.

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

Find how many times the graphs of the quadratic functions F(x) = 3x^2 - x + 7 and G(x) = x^2 - 4x + 14 intersect. This solution uses algebraic concepts such as the quadratic equation formula and the discriminant. Suitable for high school students in Grades 9-12.

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