How do we increase the roots of a quadratic equation by $1$?

How do we increase the roots of a quadratic equation by $1$?

Question

If $\alpha$ and $\beta$ are the roots of the equation $2x^2+3x+2 = 0,$ find the equation whose roots are $\alpha+1$ and $\beta+1$.

Suggestion

Can we solve this problem without computing $\alpha$ and $\beta$ explicitly?

Solution

If $y = f(x)$ has roots $\alpha$ and $\beta$, then we can find a function with roots $\alpha+1$ and $\beta+1$ by translating this $1$ unit to the right.

Thus $y = f(x-1)$ will have roots $\alpha + 1$, $\beta + 1$.

So the equation we seek is $2(x-1)^2 +3(x-1) +2 =0$, or $2x^2-x+1=0$.

It is interesting to note that this works, even though the roots of the original equation are complex!