
q(x)=−x2+6x−5q(x)=−x2+6x−5

Step 2 of 4 :  

Find the x-intercepts, if any. Express the intercept(s) as ordered pair(s).

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.

Learn how to find the x-intercepts of the quadratic function -x^2 + 6x - 5 step-by-step. Understand the quadratic formula and its application, including calculating the discriminant. Suitable for high school students.

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