The given equation is:

\[
(x - 3)^2 = 4y - 8
\]

### Step 1: Simplify the equation
First, we solve for \(y\) by isolating it on one side of the equation.

Add 8 to both sides:

\[
(x - 3)^2 + 8 = 4y
\]

Now, divide both sides by 4 to solve for \(y\):

\[
y = \frac{(x - 3)^2 + 8}{4}
\]

Simplify the expression:

\[
y = \frac{(x - 3)^2}{4} + 2
\]

### Final Answer:
The equation solved for \(y\) is:

\[
y = \frac{(x - 3)^2}{4} + 2
\]

This represents a **parabola** that opens upwards because the quadratic term \((x - 3)^2\) is positive. The vertex of this parabola is at \((3, 2)\), since the term \((x - 3)\) shifts the graph horizontally and the constant \(2\) shifts it vertically.

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Would you like further clarifications or to explore related questions? Here are some possibilities:

1. How would you convert this equation to vertex form?
2. What would the graph of this equation look like?
3. How would this equation change if it was a downward-facing parabola?
4. How do you find the focus and directrix of this parabola?
5. How does this compare to the general form of a parabola equation?

**Tip:** Parabolas in the form \((x - h)^2 = 4p(y - k)\) have a vertex at \((h, k)\) and open up or down depending on the sign of \(p\).

Discover how to solve the quadratic function (x - 3)² = 4y - 8. This problem covers algebraic concepts like parabolas, vertex form, and graph transformations. Ideal for students in Grades 9-11 learning about quadratic functions and their graphs.

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