To convert the quadratic function \( J(x) = -4x^2 + 28x + 45 \) into vertex form, we will complete the square. Here's the process:

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms.
We begin by factoring out \( -4 \) from the \( x^2 \)-term and \( x \)-term:
\[
J(x) = -4(x^2 - 7x) + 45
\]

### Step 2: Complete the square.
To complete the square, we take half of the coefficient of \( x \) inside the parentheses (which is \( -7 \)), square it, and then add and subtract it within the parentheses. 

Half of \( -7 \) is \( -\frac{7}{2} \), and squaring this gives \( \left(\frac{7}{2}\right)^2 = \frac{49}{4} \).

Now, add and subtract \( \frac{49}{4} \) inside the parentheses:
\[
J(x) = -4\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4}\right) + 45
\]

### Step 3: Simplify the expression.
Group the perfect square trinomial and factor it:
\[
J(x) = -4\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4}\right) + 45
\]

Now distribute the \( -4 \) to both terms inside the parentheses:
\[
J(x) = -4\left(x - \frac{7}{2}\right)^2 + 4 \times \frac{49}{4} + 45
\]
\[
J(x) = -4\left(x - \frac{7}{2}\right)^2 + 49 + 45
\]

### Step 4: Final expression.
Combine the constants:
\[
J(x) = -4\left(x - \frac{7}{2}\right)^2 + 94
\]

### Vertex form:
The vertex form of the quadratic is:
\[
J(x) = -4\left(x - \frac{7}{2}\right)^2 + 94
\]

The vertex of the parabola is \( \left(\frac{7}{2}, 94\right) \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you find the vertex of a parabola from its standard form?
2. What is the significance of the vertex in a quadratic function?
3. How can you determine whether a parabola opens upwards or downwards?
4. How do you use the quadratic formula to find the roots of a quadratic equation?
5. What are the effects of the coefficients in the vertex form on the graph?

**Tip:** The vertex form of a quadratic equation gives you the vertex directly, making it easy to identify the highest or lowest point on the graph.

Explore how to convert the quadratic function J(x) = -4x^2 + 28x + 45 into vertex form using the completing the square method. Learn key algebra concepts and formulas, including vertex form and identifying the vertex of a parabola. Suitable for Grades 9-12.

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