Solve the equation by graphing:

[tex]\[ -x^2 + 12x - 27 = 0 \][/tex]

First, graph the associated parabola by plotting the vertex and four additional points, two on each side. Then, use the graph to determine the solution(s) to the equation. If there is more than one solution, separate them with commas.

Solution(s): [tex]\( x = \)[/tex]

To solve the equation [tex]\( -x^2 + 12x - 27 = 0 \)[/tex] by graphing, we need to graph the associated parabola and find its roots. Let's break down the process step by step.

### Step 1: Identify the coefficients
The quadratic equation is given by [tex]\(-x^2 + 12x - 27\)[/tex]. Here, the coefficients are:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 12\)[/tex]
- [tex]\(c = -27\)[/tex]

### Step 2: Calculate the vertex of the parabola
The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]

Plugging in the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ x = -\frac{12}{2(-1)} = 6 \][/tex]

Now, find the y-coordinate of the vertex by substituting [tex]\(x = 6\)[/tex] back into the equation:
[tex]\[ y = -6^2 + 12(6) - 27 \][/tex]
[tex]\[ y = -36 + 72 - 27 \][/tex]
[tex]\[ y = 9 \][/tex]

So, the vertex is at the point [tex]\((6, 9)\)[/tex].

### Step 3: Calculate additional points on either side of the vertex
To get a good graph, we will calculate the points for [tex]\(x = 5\)[/tex], [tex]\(x = 7\)[/tex], [tex]\(x = 4\)[/tex], and [tex]\(x = 8\)[/tex].

#### For [tex]\(x = 5\)[/tex]:
[tex]\[ y = -(5)^2 + 12(5) - 27 \][/tex]
[tex]\[ y = -25 + 60 - 27 \][/tex]
[tex]\[ y = 8 \][/tex]

So, the point is [tex]\((5, 8)\)[/tex].

#### For [tex]\(x = 7\)[/tex]:
[tex]\[ y = -(7)^2 + 12(7) - 27 \][/tex]
[tex]\[ y = -49 + 84 - 27 \][/tex]
[tex]\[ y = 8 \][/tex]

So, the point is [tex]\((7, 8)\)[/tex].

#### For [tex]\(x = 4\)[/tex]:
[tex]\[ y = -(4)^2 + 12(4) - 27 \][/tex]
[tex]\[ y = -16 + 48 - 27 \][/tex]
[tex]\[ y = 5 \][/tex]

So, the point is [tex]\((4, 5)\)[/tex].

#### For [tex]\(x = 8\)[/tex]:
[tex]\[ y = -(8)^2 + 12(8) - 27 \][/tex]
[tex]\[ y = -64 + 96 - 27 \][/tex]
[tex]\[ y = 5 \][/tex]

So, the point is [tex]\((8, 5)\)[/tex].

### Step 4: Plot the points
We now have the following points to plot:
- Vertex: [tex]\((6, 9)\)[/tex]
- Points: [tex]\((5, 8)\)[/tex], [tex]\((7, 8)\)[/tex], [tex]\((4, 5)\)[/tex], [tex]\((8, 5)\)[/tex]

### Step 5: Draw the parabola and find the roots
Plot the points and draw a smooth curve through them.

### Step 6: Determine the solution(s) from the graph
The solutions to the equation [tex]\(-x^2 + 12x - 27 = 0\)[/tex] are the x-values where the parabola intersects the x-axis (where [tex]\(y = 0\)[/tex]).

From our calculations, the roots are:
- [tex]\(x = 3\)[/tex]
- [tex]\(x = 9\)[/tex]

### Solution
Hence, the solutions to the equation [tex]\(-x^2 + 12x - 27 = 0\)[/tex] are:
[tex]\[ x = 3, 9 \][/tex]

These are the points where the parabola intersects the x-axis.