To find the vertex of the parabola given by the equation [tex]\(x^2 - 14x + 4y + 29 = 0\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] terms. Here is a step-by-step solution:
1. Rearrange the equation to isolate [tex]\(4y\)[/tex]:
[tex]\[
x^2 - 14x + 4y + 29 = 0 \implies 4y = -x^2 + 14x - 29
\][/tex]
2. Move the constant term to the other side:
[tex]\[
4y = -x^2 + 14x - 29
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] terms:
We look at the expression [tex]\(-x^2 + 14x\)[/tex]. To complete the square, we rewrite this by factoring out the negative sign and then completing the square:
- First, factor out the -1 from the [tex]\(x\)[/tex] terms:
[tex]\[
4y = -(x^2 - 14x) - 29
\][/tex]
- To complete the square for [tex]\(x^2 - 14x\)[/tex], take half of the coefficient of [tex]\(x\)[/tex] (which is -14), square it, and then add and subtract this square inside the parentheses:
[tex]\[
x^2 - 14x \quad \text{(Take half of -14: } -14/2 = -7\text{)}
\][/tex]
[tex]\[
\quad \quad \text{(Square it: } (-7)^2 = 49\text{)}
\][/tex]
[tex]\[
x^2 - 14x = (x^2 - 14x + 49 - 49) = (x - 7)^2 - 49
\][/tex]
- Substitute this back into the equation:
[tex]\[
4y = -((x - 7)^2 - 49) - 29
\][/tex]
[tex]\[
4y = -(x - 7)^2 + 49 - 29
\][/tex]
[tex]\[
4y = -(x - 7)^2 + 20
\][/tex]
4. Isolate [tex]\(y\)[/tex] by dividing both sides by 4:
[tex]\[
y = -\frac{1}{4}(x - 7)^2 + 5
\][/tex]
Now, the equation in vertex form [tex]\(y = a(x - h)^2 + k\)[/tex] reveals the vertex [tex]\((h, k)\)[/tex]. Here, [tex]\(a = -\frac{1}{4}\)[/tex], [tex]\(h = 7\)[/tex], and [tex]\(k = 5\)[/tex].
Therefore, the vertex of the parabola is:
[tex]\[
(7, 5)
\][/tex]
