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To find the \( x \)-intercepts of the function \( y = -x^2 + 5x + 2 \), we set \( y = 0 \) and solve for \( x \):

\[
0 = -x^2 + 5x + 2
\]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), with:
- \( a = -1 \)
- \( b = 5 \)
- \( c = 2 \)

We use the quadratic formula to solve for \( x \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 1: Substitute the values of \( a \), \( b \), and \( c \) into the formula.
\[
x = \frac{-5 \pm \sqrt{(5)^2 - 4(-1)(2)}}{2(-1)}
\]

### Step 2: Simplify inside the square root.
\[
x = \frac{-5 \pm \sqrt{25 + 8}}{-2}
\]
\[
x = \frac{-5 \pm \sqrt{33}}{-2}
\]

### Step 3: Simplify the expression further by factoring out the negative sign.
\[
x = \frac{5 \pm \sqrt{33}}{2}
\]

So, the \( x \)-intercepts are:
\[
x = \frac{5 \pm \sqrt{33}}{2}
\]

### Answer
The correct answer is:
- **Option D: \( x = \frac{5 \pm \sqrt{33}}{2} \)**

Would you like more details on the solution process or have further questions?

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### Related Questions:
1. How do we determine the vertex of a quadratic function?
2. What is the significance of the discriminant in solving quadratic equations?
3. Can a quadratic function have complex \( x \)-intercepts?
4. How do transformations affect the graph of a quadratic function?
5. How does the value of \( a \) in \( ax^2 + bx + c \) affect the direction of the parabola?

### Quick Tip:
In the quadratic formula, if \( b^2 - 4ac > 0 \), there are two real solutions; if it equals 0, there's one real solution; if it's less than 0, the solutions are complex.

Find the x-intercepts of the graph of the function y = -x^2 + 5x + 2.

Learn how to find the x-intercepts of the quadratic function y = -x^2 + 5x + 2 using the quadratic formula. This step-by-step solution explains each part of the formula and helps students understand the process of solving quadratic equations. Suitable for students in Grades 9-11.

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