To convert the equation \( y = x^2 - 12x + 7 \) to vertex form, we will complete the square. The vertex form of a quadratic equation is:

\[
y = a(x - h)^2 + k
\]

Where \( (h, k) \) is the vertex.

### Step 1: Group the \( x \)-terms
The equation is:

\[
y = x^2 - 12x + 7
\]

Group the \( x^2 \) and \( x \) terms:

\[
y = (x^2 - 12x) + 7
\]

### Step 2: Complete the square
To complete the square, we take half of the coefficient of \( x \), square it, and add it inside the parentheses. The coefficient of \( x \) is \( -12 \), so:

\[
\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36
\]

Now, add and subtract 36 inside the parentheses:

\[
y = (x^2 - 12x + 36 - 36) + 7
\]

### Step 3: Simplify
The expression inside the parentheses becomes a perfect square trinomial:

\[
y = ((x - 6)^2 - 36) + 7
\]

Now, simplify the constant terms:

\[
y = (x - 6)^2 - 36 + 7
\]

\[
y = (x - 6)^2 - 29
\]

### Final Answer:
The vertex form of the equation is:

\[
y = (x - 6)^2 - 29
\]

The vertex of the parabola is \( (6, -29) \).

Would you like further details or have any questions about this? Here are 5 related questions:

1. How do you find the vertex of a parabola using the formula?
2. Can you convert the vertex form back to standard form?
3. How does completing the square help in solving quadratic equations?
4. What is the significance of the vertex in a quadratic function?
5. How does the value of \( a \) in vertex form affect the parabola?

**Tip:** Completing the square is a useful technique not only for converting to vertex form but also for solving quadratic equations by turning them into perfect squares.

Learn how to convert the quadratic equation y = x^2 - 12x + 7 into vertex form by completing the square. This step-by-step guide demonstrates key algebraic techniques for transforming quadratic equations and finding the vertex. Suitable for Grades 8-10 students.

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