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?

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.