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To find the vertex form of the quadratic function \( f(x) = 5x^2 - 30x + 56 \) by completing the square, let's follow these steps:

### Step 1: Factor Out the Leading Coefficient
Rewrite the expression by factoring out \( 5 \) from the terms involving \( x \):
\[
f(x) = 5(x^2 - 6x) + 56
\]

### Step 2: Complete the Square
To complete the square inside the parentheses, take half of the coefficient of \( x \), square it, and then add and subtract it inside the parentheses. The coefficient of \( x \) is \( -6 \), so:
\[
\left( \frac{-6}{2} \right)^2 = (-3)^2 = 9
\]

Add and subtract \( 9 \) inside the parentheses:
\[
f(x) = 5(x^2 - 6x + 9 - 9) + 56
\]
\[
= 5((x - 3)^2 - 9) + 56
\]

### Step 3: Distribute the Leading Coefficient
Now expand by distributing the \( 5 \):
\[
f(x) = 5(x - 3)^2 - 5 \cdot 9 + 56
\]
\[
= 5(x - 3)^2 - 45 + 56
\]

### Step 4: Simplify the Constant Terms
Combine the constant terms:
\[
f(x) = 5(x - 3)^2 + 11
\]

### Vertex Form
Now, we have the function in vertex form:
\[
f(x) = a(x - h)^2 + k
\]
where:
- \( a = 5 \)
- \( h = 3 \)
- \( k = 11 \)

### Final Answer
So, the values are:
\[
a = 5, \quad h = 3, \quad k = 11
\]

Would you like further details on any of these steps, or do you have any questions?

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Here are five related questions to deepen your understanding:

1. How does completing the square help in finding the vertex of a quadratic function?
2. Why is factoring out the leading coefficient essential in completing the square?
3. What would the vertex form look like if the function had a negative leading coefficient?
4. How can you verify that \( h = 3 \) and \( k = 11 \) represents the vertex of the function?
5. What are some alternative methods for converting a quadratic function to vertex form?

**Tip:** The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) is helpful for identifying the vertex \((h, k)\) directly, making it useful for graphing and analyzing the function’s properties.

The vertex form of the quadratic function f(x) = 5x^2 - 30x + 56 is f(x) = a(x - h)^2 + k. Determine the values for a, h, and k by completing the square.

Learn how to rewrite the quadratic function f(x) = 5x^2 - 30x + 56 in vertex form by completing the square. This step-by-step solution helps you find the values of a, h, and k, suitable for students in Grades 9-10.

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