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The given quadratic function is:

\[
f(x) = -2x^2 + 4x + 3
\]

To write this quadratic equation in **vertex form**, we need to convert it using the process of **completing the square**. The vertex form of a quadratic equation is:

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

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

### Step-by-step solution:

1. **Start with the quadratic equation:**
   \[
   f(x) = -2x^2 + 4x + 3
   \]

2. **Factor out the coefficient of \(x^2\) (which is -2) from the first two terms:**
   \[
   f(x) = -2(x^2 - 2x) + 3
   \]

3. **Complete the square:**
   To complete the square, take half of the coefficient of \(x\) (which is -2), square it, and add that inside the parentheses. The coefficient of \(x\) is -2, so:

   \[
   \left(\frac{-2}{2}\right)^2 = 1
   \]

   So, add and subtract 1 inside the parentheses:

   \[
   f(x) = -2(x^2 - 2x + 1 - 1) + 3
   \]
   \[
   f(x) = -2((x - 1)^2 - 1) + 3
   \]

4. **Simplify the expression:**
   Now, distribute the \(-2\) to both terms inside the parentheses:

   \[
   f(x) = -2(x - 1)^2 + 2 + 3
   \]
   \[
   f(x) = -2(x - 1)^2 + 5
   \]

Thus, the quadratic equation in **vertex form** is:

\[
f(x) = -2(x - 1)^2 + 5
\]

The vertex of the parabola is \((h, k) = (1, 5)\).

Would you like further details or explanations? Let me know if you have any questions!

Here are 5 related questions for further exploration:
1. How do you convert a quadratic equation from vertex form back to standard form?
2. What is the significance of the vertex in a quadratic function?
3. How does the value of "a" affect the shape of the parabola?
4. Can you find the axis of symmetry from the vertex form?
5. How would the vertex form change if the equation had no linear term (e.g., \(f(x) = -2x^2 + 3\))?

**Tip:** When completing the square, always ensure that you balance the equation by adding and subtracting the same value inside the parentheses.

Write the quadratic equation in vertex form: f(x) = -2x^2 + 4x + 3

Learn how to convert the quadratic equation f(x) = -2x^2 + 4x + 3 to vertex form using the method of completing the square. Step-by-step solution provided for students in Grades 9-12.

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