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The given quadratic equation is:

\[
y = -2x^2 + 3x - 1
\]

We are asked to rewrite this equation in vertex form, which is:

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

where \((h, k)\) is the vertex of the parabola.

### Step 1: Completing the Square
First, we'll complete the square to rewrite the quadratic in vertex form.

1. **Factor out the coefficient of \(x^2\) from the first two terms**:

\[
y = -2(x^2 - \frac{3}{2}x) - 1
\]

2. **Complete the square**:
   - Take half of the coefficient of \(x\) inside the parentheses, \(\frac{-3}{2}\), which is \(\frac{-3}{4}\).
   - Square this value: \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\).

Now add and subtract this inside the parentheses to complete the square:

\[
y = -2\left(x^2 - \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) - 1
\]

3. **Simplify**:
   - Combine the perfect square and adjust the constant outside:

\[
y = -2\left[\left(x - \frac{3}{4}\right)^2 - \frac{9}{16}\right] - 1
\]

4. **Distribute**:
   - Distribute the \(-2\) to both terms inside the brackets:

\[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{18}{16} - 1
\]

5. **Simplify constants**:
   - Simplify the constant terms:

\[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{18}{16} - \frac{16}{16}
\]

\[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{2}{16}
\]

\[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}
\]

### Step 2: Identify the vertex
Now that we have the equation in vertex form:

\[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}
\]

The vertex \((h, k)\) is \(\left(\frac{3}{4}, \frac{1}{8}\right)\).

### Final Answer:
- Vertex form: \[
y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}
\]
- Vertex: \(\left(\frac{3}{4}, \frac{1}{8}\right)\)

Would you like further explanation or details on any of the steps? 

Here are some related questions you might consider:

1. What is the axis of symmetry for this quadratic equation?
2. How does the value of \(a = -2\) affect the shape of the parabola?
3. How do you determine the direction in which the parabola opens?
4. Can you find the y-intercept of the original quadratic equation?
5. How do you find the x-intercepts of the quadratic equation?

**Tip**: Completing the square is a helpful technique not only for finding the vertex form of a quadratic equation but also for solving quadratic equations algebraically.

Write each quadratic equation in the form y = a(x - h)^2 + k. Then find its vertex. y = -2x^2 + 3x - 1

Learn how to convert the quadratic equation y = -2x^2 + 3x - 1 into vertex form and find its vertex using the method of completing the square. This step-by-step guide covers key algebraic concepts for high school students in Grades 9-11.

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