Of course! Let's convert the given function [tex]\( f(x) = 4(x + \sigma)^2 + 8 \)[/tex] into the standard quadratic form [tex]\( f(x) = ax^2 + bx + c \)[/tex].
To begin with, we need to expand and simplify the given expression step by step.
### Step 1: Expand the Squared Term
The expression inside the parentheses is [tex]\( (x + \sigma) \)[/tex]. When squared, it becomes:
[tex]\[
(x + \sigma)^2 = x^2 + 2\sigma x + \sigma^2
\][/tex]
### Step 2: Multiply by 4
Next, we multiply the entire expanded expression by 4:
[tex]\[
4(x + \sigma)^2 = 4(x^2 + 2\sigma x + \sigma^2)
\][/tex]
This simplifies to:
[tex]\[
4x^2 + 8\sigma x + 4\sigma^2
\][/tex]
### Step 3: Add the Constant Term
Now, we need to add the constant term 8 from the original function [tex]\( f(x) = 4(x + \sigma)^2 + 8 \)[/tex]:
[tex]\[
4x^2 + 8\sigma x + 4\sigma^2 + 8
\][/tex]
### Step 4: Combine Like Terms
Our function in standard quadratic form [tex]\( f(x) = ax^2 + bx + c \)[/tex] is now:
[tex]\[
f(x) = 4x^2 + 8\sigma x + (4\sigma^2 + 8)
\][/tex]
### Step 5: Identify the Coefficients
Through this expanded form, we can identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( a \)[/tex], thus [tex]\( a = 4 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( b \)[/tex], thus [tex]\( b = 8\sigma \)[/tex].
- The constant term is [tex]\( c \)[/tex], thus [tex]\( c = 4\sigma^2 + 8 \)[/tex].
So, the final coefficients are:
[tex]\[
\boxed{(a = 4, \, b = 8\sigma, \, c = 4\sigma^2 + 8)}
\][/tex]
This gives us the standard quadratic form of the given function.
