I OM - B - Task 1

Determine the coefficients $ a $ and $ b $ of the equation $ x^2 + ax + b=0 $ such that the values of $ a $ and $ b $ themselves are roots of this equation.
According to the theorems about the sum and product of the roots of a quadratic equation, the numbers $ a $ and $ b $ are the roots of the equation $ x^2 + ax + b = 0 $ if and only if they satisfy the system of equations



that is,



The second equation of the system (1) is satisfied when $ b = 0 $ or when $ a-1= 0 $, i.e., when $ a = 1 $.
System (1) therefore has two solutions:



The quadratic equations with the desired property are