Question

If the roots of the equation $x^2 + ax + b = 0$ are $c$ and $d$, then one of the roots of the equation $x^{2}+\left(2c+a\right)x+c^{2}+ac+b=0$ is

• A. $c$
• B. $d-c$
• C. $2\,d$
• D. $2\,c$

Answer 1

Answer: (B)

$d-c$

Answer 2

$f(x)=x^{2}+a x+b$, then
$f(x+c)=(x+c)^{2}+a(x+c)+b$
$=x^{2}+(2\,c+a) x+c^{2}+a c+b$
which shows that the roots of $f(x)$ are transformed to $(x-c)$ i.e., roots of $f(x+c)=0$ are $c-c$ and $d-c$.
Hence, one of the roots of the equation $f(x+c)$ is $(d-c)$.