`2x ^(2) -28x +98 =a (x -b) ^(2)`
In the expression above, `a gt 1` and both a and b are constants. Which of the following could be the value of b ?

Options

A. `-7`
B. 7
C. 14
D. 49

Explanation

To solve the equation \( 2x^2 - 28x + 98 = a(x - b)^2 \), we will follow these steps:

### Step 1: Rearranging the equation
We start with the equation:
\[ 2x^2 - 28x + 98 = a(x - b)^2 \]
We can factor out the 2 from the left side:
\[ 2(x^2 - 14x + 49) = a(x - b)^2 \]

### Step 2: Completing the square
Next, we will complete the square for the expression \( x^2 - 14x + 49 \):
\[ x^2 - 14x + 49 = (x - 7)^2 \]
Thus, we can rewrite the left side:
\[ 2((x - 7)^2) = a(x - b)^2 \]

### Step 3: Equating both sides
Now we have:
\[ 2(x - 7)^2 = a(x - b)^2 \]
To compare both sides, we can see that the coefficients of the squares must be equal. This gives us:
\[ a = 2 \]
And since both sides have the same squared term, we also have:
\[ x - 7 = x - b \]

### Step 4: Solving for b
From the equation \( x - 7 = x - b \), we can deduce:
\[ b = 7 \]

### Conclusion
Thus, the value of \( b \) is \( 7 \).