The number of ordered pairs (a, b) of integer | vedclass

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(B) Let $\alpha$ be the common real root of the equations $x^2 - ax + b = 0$ and $x^3 - ax^2 + bx + a - b = 0$.Since $\alpha$ is a root of the first e

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$2018$

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(B) Let $\alpha$ be the common real root of the equations $x^2 - ax + b = 0$ and $x^3 - ax^2 + bx + a - b = 0$.Since $\alpha$ is a root of the first equation,we have $\alpha^2 - a\alpha + b = 0$.Substituting this into the second equation: $\alpha(\alpha^2 - a\alpha + b) + a - b = 0$.Since $\alpha^2 - a\alpha + b = 0$,this simplifies to $a - b = 0$,which means $a = b$.Substituting $a = b$ into the first equation,we get $x^2 - ax + a = 0$.For the roots to be real,the discriminant $D \geq 0$,so $a^2 - 4a \geq 0$.This implies $a(a - 4) \geq 0$,which gives $a \leq 0$ or $a \geq 4$.Given $1 \leq a, b \leq 2021$ and $a = b$,we must have $4 \leq a \leq 2021$.The number of such integers $a$ is $2021 - 4 + 1 = 2018$.

The number of ordered pairs $(a, b)$ of integers such that $1 \leq a, b \leq 2021$ and the equations $x^2 - ax + b = 0$ and $x^3 - ax^2 + bx + a - b = 0$ have a common real root is

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