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

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?

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 $$2020.
So $$[\alpha]=2020$$
$$f(2022)$$ is negative and $$f(2023)$$ is positive, so $$f(x)=0$$ for some value of $$2022.
So $$[\beta]=2022$$