Find the coordinates of the vertex of the parabola y= x^2 - 18x + 74  and its intersections with the x-axis.
The vertex has coordinates x and y. find the x and y
and here are the answer options, if one of them are true, tell me and also tell me the values of "?"
The parabola and  the x-axis:
a) don't intersect 
b) have one intersection at x = ?
c) have two intersections at x1 = ? and x2 = ? .

We are given the quadratic equation of a parabola:

\[
y = x^2 - 18x + 74
\]

### Step 1: Find the coordinates of the vertex

The equation is in the standard form \(y = ax^2 + bx + c\), where \(a = 1\), \(b = -18\), and \(c = 74\). The x-coordinate of the vertex of a parabola is given by the formula:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

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

\[
x_{\text{vertex}} = \frac{-(-18)}{2(1)} = \frac{18}{2} = 9
\]

Now, substitute \(x = 9\) into the original equation to find the y-coordinate of the vertex:

\[
y_{\text{vertex}} = (9)^2 - 18(9) + 74 = 81 - 162 + 74 = -81 + 74 = -7
\]

So, the vertex is at \( (9, -7) \).

### Step 2: Find the intersections with the x-axis

The x-axis intersections occur when \(y = 0\). So we set the equation equal to 0:

\[
x^2 - 18x + 74 = 0
\]

This is a quadratic equation. To determine the number of intersections, we calculate the discriminant (\(\Delta\)):

\[
\Delta = b^2 - 4ac = (-18)^2 - 4(1)(74) = 324 - 296 = 28
\]

Since the discriminant is positive (\(\Delta > 0\)), the parabola has **two real intersections** with the x-axis. We can now find these two intersections using the quadratic formula:

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

Substitute the values of \(b\), \(a\), and \(\Delta\):

\[
x = \frac{-(-18) \pm \sqrt{28}}{2(1)} = \frac{18 \pm \sqrt{28}}{2} = \frac{18 \pm 2\sqrt{7}}{2} = 9 \pm \sqrt{7}
\]

Thus, the two intersections are:

\[
x_1 = 9 - \sqrt{7}, \quad x_2 = 9 + \sqrt{7}
\]

### Final Answer:

- The vertex is at \((9, -7)\).
- The parabola has **two intersections** with the x-axis, at \(x_1 = 9 - \sqrt{7}\) and \(x_2 = 9 + \sqrt{7}\).

Thus, the correct answer is **option (c)**: 

- Two intersections at \(x_1 = 9 - \sqrt{7}\) and \(x_2 = 9 + \sqrt{7}\).

---

Would you like further details or explanations? 

Here are 5 related questions for further exploration:
1. How does the discriminant determine the number of intersections of a parabola with the x-axis?
2. How would the vertex change if the equation had a different \(b\) value?
3. How does completing the square relate to finding the vertex of a parabola?
4. How can the quadratic formula be used in real-life applications?
5. What is the geometric significance of the vertex of a parabola?

**Tip:** Always check the discriminant before solving quadratics to determine the number of solutions!

Find the coordinates of the vertex of the parabola y = x^2 - 18x + 74 and its intersections with the x-axis. The vertex has coordinates x and y. Find the x and y, and here are the answer options: a) don't intersect, b) have one intersection at x = ?, c) have two intersections at x1 = ? and x2 = ?.

Learn how to find the vertex of the quadratic equation y = x^2 - 18x + 74 and its intersections with the x-axis. The solution includes the use of vertex formula, discriminant calculation, and quadratic formula. Suitable for high school students, Grades 9-12.

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