To find the x-intercepts of the quadratic function $q(x) = -x^2 + 6x - 5$, we need to solve the equation $q(x) = 0$. The x-intercepts are the points where the graph of the function crosses the x-axis, i.e., where $q(x) = 0$.

1. Set the function equal to zero:

$-q^2 + 6x - 5 = 0$

2. Solve the quadratic equation using the quadratic formula:

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = -1$, $b = 6$, and $c = -5$.

3. Calculate the discriminant:

$\text{Discriminant} = b^2 - 4ac = 6^2 - 4(-1)(-5) = 36 - 20 = 16$

4. Find the roots using the quadratic formula:

$x = \frac{-6 \pm \sqrt{16}}{2(-1)}$

$x = \frac{-6 \pm 4}{-2}$

This gives us two possible solutions for $x$:

$x_1 = \frac{-6 + 4}{-2} = \frac{-2}{-2} = 1$

$x_2 = \frac{-6 - 4}{-2} = \frac{-10}{-2} = 5$

5. Thus, the x-intercepts are $x = 1$ and $x = 5$.

6. Express the intercepts as ordered pairs:

The x-intercepts are $(1, 0)$ and $(5, 0)$.

Would you like more details or have any questions about this step? Here are some related questions you might find useful:

1. How do you determine the number of x-intercepts a quadratic function has?
2. What does the discriminant tell you about the roots of a quadratic equation?
3. How do you find the y-intercept of a quadratic function?
4. What are the real-world interpretations of x-intercepts in applications?
5. Can a quadratic function have no x-intercepts? If so, what does that mean?

Tip: The discriminant ($b^2 - 4ac$) determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are no real roots, only complex ones.