To find the [tex]\( x \)[/tex]-intercepts of the parabola defined by the equation [tex]\( y = x^2 - 9x + 18 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].
1. Set the quadratic equation equal to zero:
[tex]\[
x^2 - 9x + 18 = 0
\][/tex]
2. To solve the quadratic equation, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
3. For our equation [tex]\( x^2 - 9x + 18 = 0 \)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = -9, \quad c = 18
\][/tex]
4. First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = (-9)^2 - 4 \cdot 1 \cdot 18 = 81 - 72 = 9
\][/tex]
5. The quadratic formula now gives us:
[tex]\[
x = \frac{-(-9) \pm \sqrt{9}}{2 \cdot 1} = \frac{9 \pm 3}{2}
\][/tex]
6. Solving for the two roots:
[tex]\[
x_1 = \frac{9 + 3}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x_2 = \frac{9 - 3}{2} = \frac{6}{2} = 3
\][/tex]
7. Therefore, the [tex]\( x \)[/tex]-intercepts of the parabola are at [tex]\( (6, 0) \)[/tex] and [tex]\( (3, 0) \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{(3,0) \text{ and } (6,0)} \][/tex]
Therefore, the correct option is [tex]\( \text{D} \)[/tex].
