Complete the square
to find the vertex
of this parabola.

Ask by Alexander Johnston. in the United States

Jan 20,2025

Question

Complete the square to find the vertex of this parabola. $$ y^{2}+12 x-10 y+1=0 $$ $$ ([?],[]) $$

UpStudy AI · Accepted Answer

To find the vertex of the parabola defined by the equation:

$$ y^{2} + 12x - 10y + 1 = 0 $$

we will **complete the square** for the $$ y $$-terms.

### Step 1: Rearrange the Equation
First, let's group the $$ y $$-terms and move the other terms to the opposite side:

$$ y^{2} - 10y = -12x - 1 $$

### Step 2: Complete the Square for $$ y $$
To complete the square for the expression $$ y^{2} - 10y $$:

1. **Find the coefficient of $$ y $$**: Here, it's $$-10$$.
2. **Take half of this coefficient**: $$-10 / 2 = -5$$.
3. **Square this value**: $$(-5)^2 = 25$$.

Add and subtract this square inside the equation:

$$ y^{2} - 10y + 25 = -12x - 1 + 25 $$

This simplifies to:

$$ (y - 5)^2 = -12x + 24 $$

### Step 3: Rewrite in Vertex Form
The equation now resembles the standard vertex form of a horizontal parabola:

$$ (y - k)^2 = 4p(x - h) $$

Comparing both equations:

$$ (y - 5)^2 = -12(x - 2) $$

Here:
- $$ h = 2 $$
- $$ k = 5 $$
- $$ 4p = -12 $$ (which indicates the parabola opens to the left)

### Step 4: Identify the Vertex
From the vertex form, the vertex $$(h, k)$$ is:

$$ \boxed{(2,\ 5)} $$
The vertex of the parabola is at $$(2, 5)$$.

Complete the square
to find the vertex
of this parabola.

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