Answer :
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.
### 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.
