Question

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

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

Answer 1

The correct option is B

$b=4,c=-1$

Explanation for the correct answer.

Given:$f(x+1)-f\left(x\right)=8x+3$, $f\left(x\right)=b{x}^2+cx+d$

$\begin{array}{rrcc}\Rightarrow & f(x+1)-f\left(x\right)& =& 8x+3\\ \Rightarrow & b{(x+1)}^2+c(x+1)+d-b{x}^2-cx-d& =& 8x+3\\ \Rightarrow & b\left[{(x+1)}^2-x^2\right]+c& =& 8x+3\\ \Rightarrow & b\left[x^2+1+2x-x^2\right]+c& =& 8x+3\\ \Rightarrow & b\left[1+2x\right]+c& =& 8x+3\\ \Rightarrow & b+c+2bx& =& 8x+3\end{array}$

Now compare like terms

$$\begin{gathered} \Rightarrow 2\mathrm{b}=8 \\ \therefore \mathrm{b}=4 \end{gathered}$$

$$\begin{gathered} \Rightarrow \mathrm{b}+\mathrm{c}=3 \\ \Rightarrow 4+\mathrm{c}=3 \\ \Rightarrow \mathrm{c}=3-4 \\ \therefore \mathrm{c}=-1 \end{gathered}$$

Hence option (B) is the correct option.