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The problem states the quadratic equation \( y = x^2 - 2x - 1 \), and asks to write it in **vertex form**.

### Vertex Form of a Quadratic Equation:
The vertex form is written as:
\[
y = a(x-h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.

To convert \( y = x^2 - 2x - 1 \) into vertex form, we use **completing the square**:

---

### Step 1: Group the quadratic and linear terms
\[
y = (x^2 - 2x) - 1
\]

---

### Step 2: Complete the square
Take half the coefficient of \(x\), square it, and add/subtract it inside the parentheses.

The coefficient of \(x\) is \(-2\). Half of \(-2\) is \(-1\), and squaring it gives:
\[
(-1)^2 = 1
\]

Now add and subtract \(1\) inside the parentheses:
\[
y = (x^2 - 2x + 1 - 1) - 1
\]

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

---

### Step 3: Simplify
\[
y = (x - 1)^2 - 2
\]

---

### Final Answer:
The equation in vertex form is:
\[
y = (x - 1)^2 - 2
\]

The vertex is at \((1, -2)\).

---

Would you like further clarifications or additional examples?  

### Related Questions:
1. How do you find the vertex of a parabola given in standard form?
2. What is the axis of symmetry of the parabola \( y = (x - 1)^2 - 2 \)?
3. How does completing the square relate to factoring quadratics?
4. What are some practical applications of the vertex form of a quadratic equation?
5. How do you graph a parabola given in vertex form?

### Tip:
When completing the square, always check your work by expanding the vertex form back into standard form to ensure correctness!

Write the equation y = x^2 - 2x - 1 in vertex form.

Learn how to rewrite the quadratic equation y = x^2 - 2x - 1 in vertex form by completing the square. Step-by-step guidance for algebra students in Grades 8-10, including key concepts and formulas.

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