convert standard form of parabola to general form (y+2)^2=8(x-1)

To convert the given standard form of a parabola into its general form, we'll expand and rearrange the equation. The given equation is:

\[
(y + 2)^2 = 8(x - 1)
\]

### Step 1: Expand the equation

Expand the left-hand side:

\[
(y + 2)(y + 2) = y^2 + 4y + 4
\]

So, the equation becomes:

\[
y^2 + 4y + 4 = 8(x - 1)
\]

### Step 2: Distribute on the right-hand side

Expand the right-hand side:

\[
8(x - 1) = 8x - 8
\]

So, the equation is now:

\[
y^2 + 4y + 4 = 8x - 8
\]

### Step 3: Rearrange into general form

To convert this into the general form of a quadratic equation, bring all the terms to one side:

\[
y^2 + 4y - 8x + 12 = 0
\]

This is the general form of the given parabola.

### Summary
The general form of the parabola is:

\[
y^2 + 4y - 8x + 12 = 0
\]

Would you like further details or have any questions about this conversion? Here are some related questions you might find interesting:

1. How do you convert the general form of a parabola to its standard form?
2. What are the vertex and focus of the parabola \((y + 2)^2 = 8(x - 1)\)?
3. How can you identify the direction in which a parabola opens based on its equation?
4. What are the characteristics of a parabola when the equation is given in general form?
5. How do you find the axis of symmetry for the parabola in general form?

**Tip:** When converting between forms of a parabola, ensure that you correctly expand and combine like terms to avoid errors.

Learn how to convert the standard form of a parabola (y+2)^2=8(x-1) to its general form y^2 + 4y - 8x + 12 = 0 with a detailed step-by-step solution. This problem covers quadratic equations and the process of expanding and rearranging equations. Suitable for students in Grades 10-12.

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