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To convert the quadratic equation \( y = x^2 - 2x - 3 \) into the form \( y = (x - h)^2 + k \), we need to **complete the square**. Here's how to do it step-by-step:

### Step 1: Rewrite the equation
We begin with the equation:
\[
y = x^2 - 2x - 3
\]
Focus on the quadratic and linear terms: \( x^2 - 2x \).

### Step 2: Complete the square
To complete the square, take half of the coefficient of \(x\) (which is -2), square it, and add and subtract that value inside the equation. 

\[
\text{Half of } -2 \text{ is } -1, \quad (-1)^2 = 1
\]

So, we add and subtract 1:
\[
y = (x^2 - 2x + 1) - 1 - 3
\]
\[
y = (x - 1)^2 - 4
\]

### Step 3: Write in vertex form
Now the equation is in the form:
\[
y = (x - 1)^2 - 4
\]

Thus, the answer is:
\[
y = (x - 1)^2 - 4
\]

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How do you determine the vertex of a quadratic function in vertex form?
2. How does completing the square help in solving quadratic equations?
3. Can you graph the equation \(y = (x - 1)^2 - 4\) and explain the transformations?
4. How would you convert a quadratic equation from standard form to factored form?
5. What are the applications of vertex form in real-world problems?

**Tip:** When completing the square, always remember to balance the equation by adding and subtracting the same value within the expression.

Put the equation y = x^2 - 2x - 3 into the form y = (x - h)^2 + k.

Learn how to convert the quadratic equation y = x^2 - 2x - 3 into vertex form using the completing the square method. This step-by-step guide covers essential algebra concepts, including completing the square and understanding vertex form. Suitable for students in Grades 9-11.

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