Find the [tex]\( x \)[/tex]-intercepts for the parabola defined by the equation:

[tex]\[ y = x^2 - 9x + 18 \][/tex]

A. [tex]\((-3,0)\)[/tex] and [tex]\((-6,0)\)[/tex]
B. [tex]\((0,-3)\)[/tex] and [tex]\((0,-6)\)[/tex]
C. [tex]\((0,3)\)[/tex] and [tex]\((0,6)\)[/tex]
D. [tex]\((3,0)\)[/tex] and [tex]\((6,0)\)[/tex]



Answer :

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].