If `f(x) =ax^(2) + bx + c` satisfies the identity `f(x+1) -f(x)= 8x+ 3` for all `x in R` Then (a,b)=

Options

A. `(2,1)`
B. `(4,-1)`
C. `(-1,4)`
D. `(-1,1)`

Explanation

To solve the problem, we need to find the values of \(a\) and \(b\) in the function \(f(x) = ax^2 + bx + c\) given that it satisfies the identity \(f(x+1) - f(x) = 8x + 3\).

### Step-by-Step Solution:

1. **Define the Function**:
We start with the function:
\[ f(x) = ax^2 + bx + c \]

2. **Calculate \(f(x+1)\)**:
To find \(f(x+1)\), we substitute \(x\) with \(x+1\):
\[ f(x+1) = a(x+1)^2 + b(x+1) + c \]
Expanding this:
\[ f(x+1) = a(x^2 + 2x + 1) + b(x + 1) + c \]
\[ = ax^2 + 2ax + a + bx + b + c \]
\[ = ax^2 + (2a + b)x + (a + b + c) \]

3. **Subtract \(f(x)\) from \(f(x+1)\)**:
Now we compute \(f(x+1) - f(x)\):
\[ f(x+1) - f(x) = \left(ax^2 + (2a + b)x + (a + b + c)\right) - \left(ax^2 + bx + c\right) \]
Simplifying this gives:
\[ = (2a + b - b)x + (a + b + c - c) \]
\[ = 2ax + (a + b) \]

4. **Set the Expression Equal to the Given Identity**:
According to the problem, we have:
\[ f(x+1) - f(x) = 8x + 3 \]
Therefore, we can equate:
\[ 2ax + (a + b) = 8x + 3 \]

5. **Compare Coefficients**:
From the equation \(2ax + (a + b) = 8x + 3\), we can compare the coefficients of \(x\) and the constant terms:
- For the coefficient of \(x\):
\[ 2a = 8 \implies a = \frac{8}{2} = 4 \]
- For the constant term:
\[ a + b = 3 \]

6. **Substitute \(a\) to Find \(b\)**:
Now substituting \(a = 4\) into the equation \(a + b = 3\):
\[ 4 + b = 3 \implies b = 3 - 4 = -1 \]

7. **Final Result**:
Thus, we have:
\[ (a, b) = (4, -1) \]

### Summary:
The values of \(a\) and \(b\) are:
\[ (a, b) = (4, -1) \]