Find the x intercept(s) and the coordinates of the vertex for the parabola y=x²+14x+9 . If there is more than one x intercept, separate them with commas.

To find the x-intercepts and the coordinates of the vertex for the parabola \( y = x^2 + 14x + 9 \), we can follow these steps:

### 1. Finding the x-Intercepts:
The x-intercepts occur where \( y = 0 \). So, set the equation equal to zero and solve for \( x \):

\[
x^2 + 14x + 9 = 0
\]

To solve this quadratic equation, we can use the quadratic formula:

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

For the equation \( x^2 + 14x + 9 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 14 \)
- \( c = 9 \)

Plugging these values into the quadratic formula:

\[
x = \frac{-14 \pm \sqrt{14^2 - 4(1)(9)}}{2(1)}
\]
\[
x = \frac{-14 \pm \sqrt{196 - 36}}{2}
\]
\[
x = \frac{-14 \pm \sqrt{160}}{2}
\]
\[
x = \frac{-14 \pm \sqrt{16 \times 10}}{2}
\]
\[
x = \frac{-14 \pm 4\sqrt{10}}{2}
\]
\[
x = -7 \pm 2\sqrt{10}
\]

So, the x-intercepts are:

\[
x = -7 + 2\sqrt{10}, \quad x = -7 - 2\sqrt{10}
\]

### 2. Finding the Vertex:
The vertex of a parabola in the form \( y = ax^2 + bx + c \) is found using the formula for the x-coordinate of the vertex:

\[
x = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{14}{2(1)} = -7
\]

Now, substitute \( x = -7 \) back into the original equation to find the y-coordinate of the vertex:

\[
y = (-7)^2 + 14(-7) + 9
\]
\[
y = 49 - 98 + 9
\]
\[
y = -40
\]

Thus, the vertex of the parabola is at:

\[
(-7, -40)
\]

### Final Answers:
- **x-intercepts:** \( x = -7 + 2\sqrt{10}, \, x = -7 - 2\sqrt{10} \)
- **Vertex:** \( (-7, -40) \)

Would you like more details or have any other questions?

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**Related Questions:**

1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you determine the axis of symmetry of a parabola?
4. Can the quadratic formula be used for non-real solutions?
5. What are the roots of the equation \( y = x^2 + 14x + 9 \) if \( y = 0 \)?

**Tip:** The vertex form of a parabola is a useful way to quickly identify the vertex and can be derived by completing the square on the quadratic equation.

Find the x intercept(s) and the coordinates of the vertex for the parabola y=x²+14x+9. If there is more than one x intercept, separate them with commas.

Explore how to find the x-intercepts and vertex of the quadratic equation y = x² + 14x + 9. This step-by-step guide covers key algebraic concepts such as the quadratic formula and vertex calculation, suitable for students in Grades 9-11.

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