Which of the following functions is equivalent to thte function above ?

Options

A. `f (x) = (x-5) ^(2)`
B. `f (x) = x ^(2) + 10.28x + 5.42`
C. `f (x) = 0.61x ^(2) + 0.14x +25`
D. `f (x) =1.3 (x -3)^(2) =0.69x ^(2) + 0.14x +9.79`

Explanation

To solve the given function \( f(x) = (1.3x - 3.9)^2 - 0.69x^2 - 0.14x - 9.79 \) and find an equivalent function, we will follow these steps:

### Step 1: Expand the squared term
We start by expanding \( (1.3x - 3.9)^2 \) using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).

\[ (1.3x - 3.9)^2 = (1.3x)^2 - 2 \cdot (1.3x) \cdot (3.9) + (3.9)^2 \]

Calculating each term:
- \( (1.3x)^2 = 1.69x^2 \)
- \( -2 \cdot (1.3x) \cdot (3.9) = -10.14x \)
- \( (3.9)^2 = 15.21 \)

Thus, we have:

\[ (1.3x - 3.9)^2 = 1.69x^2 - 10.14x + 15.21 \]

### Step 2: Substitute back into the function
Now, substitute this expansion back into the function:

\[ f(x) = 1.69x^2 - 10.14x + 15.21 - 0.69x^2 - 0.14x - 9.79 \]

### Step 3: Combine like terms
Now, we will combine the like terms:

1. For \( x^2 \) terms:
\[ 1.69x^2 - 0.69x^2 = 1.00x^2 \quad \text{(or simply } x^2\text{)} \]

2. For \( x \) terms:
\[ -10.14x - 0.14x = -10.28x \]

3. For constant terms:
\[ 15.21 - 9.79 = 5.42 \]

Thus, we can rewrite the function as:

\[ f(x) = x^2 - 10.28x + 5.42 \]

### Step 4: Rewrite in completed square form
Next, we will rewrite this quadratic in the completed square form:

\[ f(x) = x^2 - 10.28x + 5.42 \]

To complete the square, we take half of the coefficient of \( x \) (which is -10.28), square it, and add and subtract it:

\[ \left(\frac{-10.28}{2}\right)^2 = (-5.14)^2 = 26.4196 \]

Thus, we rewrite the function:

\[ f(x) = (x^2 - 10.28x + 26.4196) - 26.4196 + 5.42 \]

This simplifies to:

\[ f(x) = (x - 5.14)^2 - 20.9996 \]

### Step 5: Identify equivalent function
The function \( f(x) = (x - 5.14)^2 - 20.9996 \) is equivalent to the original function.

### Summary
The equivalent function can be expressed in the form of a completed square. The final equivalent function is:

\[ f(x) = (x - 5.14)^2 - 20.9996 \]