Let's find the [tex]\(x\)[/tex]-intercepts of the parabola with vertex [tex]\((4, 75)\)[/tex] and [tex]\(y\)[/tex]-intercept [tex]\((0, 27)\)[/tex].
### Step-by-Step Solution
1. Identify given information:
- Vertex [tex]\((h, k) = (4, 75)\)[/tex]
- [tex]\(y\)[/tex]-intercept is the point [tex]\((0, 27)\)[/tex].
2. Write the equation of the parabola in vertex form:
The general form of a parabola with vertex [tex]\((h, k)\)[/tex] is:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
Substituting [tex]\(h = 4\)[/tex] and [tex]\(k = 75\)[/tex]:
[tex]\[
y = a(x - 4)^2 + 75
\][/tex]
3. Use the [tex]\(y\)[/tex]-intercept to find [tex]\(a\)[/tex]:
The [tex]\(y\)[/tex]-intercept is the point [tex]\((0, 27)\)[/tex]. Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 27\)[/tex] into the equation:
[tex]\[
27 = a(0 - 4)^2 + 75
\][/tex]
Simplify:
[tex]\[
27 = 16a + 75
\][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[
27 - 75 = 16a \quad \Rightarrow \quad -48 = 16a \quad \Rightarrow \quad a = -3
\][/tex]
4. Write the complete quadratic equation:
Now that we have [tex]\(a = -3\)[/tex], the quadratic equation is:
[tex]\[
y = -3(x - 4)^2 + 75
\][/tex]
5. Find the [tex]\(x\)[/tex]-intercepts by setting [tex]\(y = 0\)[/tex]:
[tex]\[
0 = -3(x - 4)^2 + 75
\][/tex]
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[
-3(x - 4)^2 = -75 \quad \Rightarrow \quad (x - 4)^2 = 25
\][/tex]
Solve for [tex]\(x\)[/tex] by taking the square root of both sides:
[tex]\[
x - 4 = \pm 5
\][/tex]
Therefore, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x - 4 = 5 \quad \Rightarrow \quad x = 9
\][/tex]
and
[tex]\[
x - 4 = -5 \quad \Rightarrow \quad x = -1
\][/tex]
6. Write the coordinates of the [tex]\(x\)[/tex]-intercepts:
The [tex]\(x\)[/tex]-intercepts are the points where the parabola crosses the [tex]\(x\)[/tex]-axis. These points are:
[tex]\[
(9, 0) \quad \text{and} \quad (-1, 0).
\][/tex]
### Final Answer
The [tex]\(x\)[/tex]-intercepts of the parabola are:
[tex]\(\left(-1, 0\right), \left(9, 0\right)\)[/tex]
Hence, the correct answer is:
[tex]$((-1, 0), (9, 0))$[/tex]
