Certainly! Let's find the vertex of the given parabola by completing the square. Here is the step-by-step solution:
### Given Equation:
[tex]\[ x^2 + 8y + 2x - 23 = 0 \][/tex]
### Step 1: Rearrange terms to group the [tex]\(x\)[/tex] terms together.
[tex]\[ x^2 + 2x + 8y - 23 = 0 \][/tex]
### Step 2: Isolate the [tex]\(y\)[/tex] term.
[tex]\[ 8y = -(x^2 + 2x) + 23 \][/tex]
[tex]\[ y = -\frac{x^2 + 2x}{8} + \frac{23}{8} \][/tex]
### Step 3: Complete the square inside the parentheses for [tex]\(x\)[/tex].
To complete the square, we take the [tex]\(x^2 + 2x\)[/tex] term.
First, find the term to complete the square:
[tex]\[ x^2 + 2x \][/tex]
To complete the square, add and subtract [tex]\((\frac{2}{2})^2 = 1\)[/tex]:
[tex]\[ x^2 + 2x + 1 - 1 = (x + 1)^2 - 1 \][/tex]
### Step 4: Substitute the completed square back into the expression for [tex]\(y\)[/tex].
[tex]\[ y = -\frac{(x + 1)^2 - 1}{8} + \frac{23}{8} \][/tex]
### Step 5: Simplify the expression.
Distribute the [tex]\(-\frac{1}{8}\)[/tex]:
[tex]\[ y = -\frac{(x + 1)^2}{8} + \frac{1}{8} + \frac{23}{8} \][/tex]
Combine the constants:
[tex]\[ y = -\frac{(x + 1)^2}{8} + \frac{24}{8} \][/tex]
[tex]\[ y = -\frac{(x + 1)^2}{8} + 3 \][/tex]
### Vertex Form of the Parabola:
[tex]\[ y = -\frac{1}{8}(x + 1)^2 + 3 \][/tex]
In vertex form [tex]\((x-h)^2\)[/tex], the vertex [tex]\((h, k)\)[/tex] is clearly visible.
### Step 6: Identify the vertex.
The vertex form tells us the vertex [tex]\((h, k)\)[/tex] is:
[tex]\( (h, k) = (-1, 3) \)[/tex]
### Conclusion:
The vertex of the parabola [tex]\( x^2 + 8y + 2x - 23 = 0 \)[/tex] is at:
[tex]\[ \boxed{(-1, 3)} \][/tex]
