Question

The values of $b$ and $c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied, where $f(x) = bx^2 + cx + d$, are

• A. b = 2, c = 1
• B. b = 4, c = -1
• C. b = 1, c = 2
• D. b = 3, c = -1

Hint:

Compare coefficients of $x$ and the constant term to find unknowns in an identity.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We substitute the general form of $f(x)$ into the identity and equate the coefficients of like powers of $x$.
Step 2: Key Formula or Approach:
$f(x+1) = b(x+1)^2 + c(x+1) + d$.
Difference $f(x+1) - f(x) = b(x^2 + 2x + 1) + c(x+1) + d - (bx^2 + cx + d)$.
Step 3: Detailed Explanation:
Simplifying the difference:
\[ bx^2 + 2bx + b + cx + c + d - bx^2 - cx - d = 2bx + (b + c) \] We are given $2bx + (b + c) = 8x + 3$.
Equating coeff of $x$: $2b = 8 \implies b = 4$.
Equating constants: $b + c = 3 \implies 4 + c = 3 \implies c = -1$.
Step 4: Final Answer:
The values are $b = 4, c = -1$.