To solve the equation \((x-3)^2 = 12(y-1)\), we can follow these steps:

### Step 1: Expand the left-hand side.
We will leave the right-hand side as it is for now:

\[
(x-3)^2 = 12(y-1)
\]

Expanding \((x-3)^2\):

\[
x^2 - 6x + 9 = 12(y-1)
\]

### Step 2: Distribute on the right-hand side.
Now distribute the \(12\) on the right-hand side:

\[
x^2 - 6x + 9 = 12y - 12
\]

### Step 3: Rearrange the equation.
Move all terms to one side to get the equation in a more standard form:

\[
x^2 - 6x - 12y + 21 = 0
\]

This is the equation of a conic section (a parabola) in general form. It can also be written as:

\[
x^2 - 6x = 12y - 21
\]

You can complete the square or leave it in this form to identify properties of the conic.

Would you like further details on how to interpret this result or any additional steps?

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Here are 5 related questions:
1. How can you graph this equation?
2. What is the vertex of this parabola?
3. Can you rewrite the equation in vertex form?
4. How can we determine if the parabola opens upwards or downwards?
5. How would this equation change if it was rotated?

**Tip:** Always try to simplify your equation into a standard form, like the vertex or general form for conic sections, for easier analysis.

Explore the solution to the equation (x-3)^2 = 12(y-1) and learn about the properties of parabolas, including completing the square and rewriting in standard form. Ideal for students in Grades 10-12 studying conic sections in Algebra.

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