turn into standard form of a parabola x^2+2y+66=0

To convert the equation \(x^2 + 2y + 66 = 0\) into the standard form of a parabola, follow these steps:

### Given Equation:
\[
x^2 + 2y + 66 = 0
\]

### Step 1: Isolate the \(y\)-term
Move the constant and \(2y\) to the other side of the equation:
\[
x^2 = -2y - 66
\]

### Step 2: Complete the square for the \(y\)-term (if necessary)
To write the equation in standard form, isolate \(y\) on one side:
\[
x^2 = -2(y + 33)
\]

### Step 3: Write the equation in standard form
The standard form of a parabola that opens either up or down is:
\[
(x - h)^2 = 4p(y - k)
\]
Comparing with \(x^2 = -2(y + 33)\), we see that:

- \(h = 0\)
- \(k = -33\)
- \(4p = -2\) so \(p = -\frac{1}{2}\)

### Final Standard Form:
\[
(x - 0)^2 = -2(y + 33)
\]
Or simply:
\[
x^2 = -2(y + 33)
\]

This is the standard form of the given parabola.

Do you want more details or have any questions?

Here are some related questions:

1. How do you determine the value of \(p\) in the standard form of a parabola?
2. What does the value of \(p\) indicate about the direction in which the parabola opens?
3. How can you determine the vertex of the parabola from the equation?
4. How would the equation change if the parabola opened horizontally instead?
5. What other forms can a parabola equation take?

**Tip**: When converting equations to standard form, always check if the equation can be rearranged to match the form \((x - h)^2 = 4p(y - k)\) or \((y - k)^2 = 4p(x - h)\).

Learn how to convert the equation x^2 + 2y + 66 = 0 into the standard form of a parabola. Follow detailed steps including isolating the y-term and completing the square, with a focus on quadratic equations and standard parabola forms. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation