Question

The three solutions of the equation $f\left(x\right)=0$ are $-2,0$ and $3$. Therefore, the three solutions of the equation $f(x-2)=0$ are

• A. $-4,-2$ and $1$
• B. $-2,0$ and $3$
• C. $4,2$ and $5$
• D. $0,2$ and $5$

Answer 1

The correct option is D $0,2$ and $5$

$f\left(x\right)$ can be written as $f\left(x\right)=(x-0)(x+2)(x-3)=x(x+2)(x-3)$. Replacing $x$ by $(x-2)$ we get $f(x-2)=(x-2)\left(x\right)(x-5)$. Hence $f(x-2)=0$ implies $x=2,0,5$ are the roots.
In other words the graph of $f(x-2)$ is the same as $f\left(x\right)$ only it has moved two units to the right. Thus the new roots will be existing roots + 2.