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Roots of quadratic satisfy equation

Source: 2007 Bulgarian Autumn Math Competition, Problem 10.1

March 17, 2022

quadraticsalgebraroots of the equation

Problem Statement

Find all integers

b

and

c

for which the equation

x^2-bx+c=0

has two real roots

x_{1}

and

x_{2}

satisfying

x_{1}^2+x_{2}^2=5