Question

The value of $b$ and $c$ for which the identify $f(x+1)-f\left(x\right)=8x+3$ 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

Answer: (B)

$b=4,c=-1$

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

Substitute $x=0$ we get $f(0+1)-f\left(0\right)=3$

$\Rightarrow (b+c+d)-d=3\Rightarrow b+c=3...\left(i\right)$

Again substitute $x=-1$

we get

$f(-1+1)-f(-1)=-8+3\Rightarrow f\left(0\right)-f(-1)=-5\Rightarrow d-(b-c+d)=-5\Rightarrow -b+c=-5......\left(ii\right)$

Solving (i) and (ii) we get $b=4,c=-1$