Which equation is y = –6x2 + 3x + 2 rewritten in vertex form?
y = negative 6 (x minus 1) squared + 8
y = negative 6 (x + one-fourth) squared + thirteen-eighths
y = negative 6 (x minus one-fourth) squared + nineteen-eighths
y = negative 6 (x minus one-half) squared + seven-halves

To rewrite the quadratic equation \( y = -6x^2 + 3x + 2 \) in vertex form, we need to complete the square.

### Steps to rewrite in vertex form:

1. **Start with the given equation**:  
   \[
   y = -6x^2 + 3x + 2
   \]

2. **Factor out \(-6\) from the first two terms**:  
   \[
   y = -6(x^2 - \frac{1}{2}x) + 2
   \]

3. **Complete the square**:  
   To complete the square, take half of the coefficient of \(x\) inside the parentheses (which is \(-\frac{1}{2}\)), square it, and add and subtract this value inside the parentheses:
   \[
   \left(\frac{-\frac{1}{2}}{2}\right)^2 = \left(-\frac{1}{4}\right)^2 = \frac{1}{16}
   \]
   So we add and subtract \(\frac{1}{16}\) inside the parentheses:
   \[
   y = -6\left(x^2 - \frac{1}{2}x + \frac{1}{16} - \frac{1}{16}\right) + 2
   \]

4. **Rewrite the perfect square**:
   \[
   y = -6\left(\left(x - \frac{1}{4}\right)^2 - \frac{1}{16}\right) + 2
   \]

5. **Simplify the equation**:
   Now, distribute the \(-6\):
   \[
   y = -6(x - \frac{1}{4})^2 + \frac{6}{16} + 2
   \]
   \[
   y = -6(x - \frac{1}{4})^2 + \frac{3}{8} + 2
   \]
   \[
   y = -6(x - \frac{1}{4})^2 + \frac{3}{8} + \frac{16}{8}
   \]
   \[
   y = -6(x - \frac{1}{4})^2 + \frac{19}{8}
   \]

Thus, the vertex form of the equation is:
\[
y = -6\left(x - \frac{1}{4}\right)^2 + \frac{19}{8}
\]

The correct choice is:
\[
\boxed{y = -6\left(x - \frac{1}{4}\right)^2 + \frac{19}{8}}
\]

---

Would you like further details or have any questions?

Here are five related questions for deeper understanding:
1. How do you complete the square in a general quadratic equation?
2. What is the vertex of the quadratic equation in vertex form?
3. How does the value of \(a\) affect the graph of a quadratic function in vertex form?
4. Can you convert the vertex form back into standard form?
5. What are the benefits of expressing a quadratic equation in vertex form?

**Tip**: The vertex form of a quadratic equation is useful because it easily reveals the vertex \((h, k)\) of the parabola, making it easier to graph and analyze.

Which equation is y = –6x^2 + 3x + 2 rewritten in vertex form? y = negative 6 (x minus 1) squared + 8, y = negative 6 (x + one-fourth) squared + thirteen-eighths, y = negative 6 (x minus one-fourth) squared + nineteen-eighths, y = negative 6 (x minus one-half) squared + seven-halves

Learn how to rewrite the quadratic equation y = –6x^2 + 3x + 2 into vertex form by completing the square. This solution explains key algebraic concepts, including completing the square and vertex form transformation. Suitable for students in Grades 9-12.

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