To determine the number of $x$-intercepts of the function $f(x) = -x^2 + 7x + 5$, we need to find the points where the graph intersects the $x$-axis. These points occur when $f(x) = 0$, i.e., when:

$-x^2 + 7x + 5 = 0$

This is a quadratic equation, which we can solve using the discriminant method. For a quadratic equation of the form $ax^2 + bx + c = 0$, the discriminant is given by:

$\Delta = b^2 - 4ac$

For the equation $-x^2 + 7x + 5 = 0$, the coefficients are:

• $a = -1$
• $b = 7$
• $c = 5$

Now, calculate the discriminant:

$\Delta = (7)^2 - 4(-1)(5) = 49 + 20 = 69$

Interpretation of the discriminant:

• If $\Delta > 0$, there are two distinct real roots, meaning the graph intersects the $x$-axis at two points.
• If $\Delta = 0$, there is one real root (the vertex touches the $x$-axis).
• If $\Delta < 0$, there are no real roots, meaning the graph does not intersect the $x$-axis.

Since $\Delta = 69 > 0$, there are two distinct real roots, which means the graph has two x-intercepts.

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Would you like to see the exact points of intersection? Or would you like to explore other aspects of the quadratic function?

Here are some related questions:

1. How do you find the exact x-intercepts of a quadratic function?
2. What does the discriminant tell you about the shape of a quadratic graph?
3. How do you determine whether a quadratic function opens upwards or downwards?
4. What is the vertex of the quadratic function $f(x) = -x^2 + 7x + 5$?
5. How can you find the axis of symmetry for a quadratic function?

Tip: The discriminant is very useful for quickly determining the number of real roots of a quadratic equation without having to fully solve it.