If $\alpha,\beta$ are roots of the equation $5(x-2020)(x-2022)+7(x-2021)(x-2023)=0$, then find $[\alpha]+[\beta]$

Question

If $\alpha,\beta$ are roots of the equation $5(x-2020)(x-2022)+7(x-2021)(x-2023)=0$, then find $[\alpha]+[\beta]$, where $[.]$ represents the greatest integer function.

I put $x-2020=t$ and got $12t^2-38t+21=0$ and got approximate roots as $2.5$ and $0.7$

So, I got $[\alpha]+[\beta]=4042$, which is correct.

I wonder if we can find the answer without calculating the roots.

Let $a,b$ be the roots of the quadratic in $t$. So, $a+b=\frac{19}{6}=3.2$(approx.)

$a=[a]+\{a\}$, where $\{a\}$ is the fractional part of $a$.

So, $[a]+\{a\}+[b]+\{b\}=3.2$

$3.2$ can be split as $2+1.2$ or $3+0.2$

Without calculating the roots, can we confidently split $3.2$ to get the desired answer?

Answer 1

Let $f(x)=5(x-2020)(x-2022)+7(x-2021)(x-2023)=0$ with roots $\alpha< \beta$.

$f(2020)$ is positive and $f(2021)$ is negative, so $f(x)=0$ for some value of $2020<x<2021$.
So $[\alpha]=2020$

$f(2022)$ is negative and $f(2023)$ is positive, so $f(x)=0$ for some value of $2022<x<2023$.
So $[\beta]=2022$