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x + y = 2 and xy = z^2 + 1, x^2 + y^2+ z^2 =?

Source: 1998 Estonia National Olympiad Final Round grade 9 p4

March 14, 2020

system of equationsalgebra

Problem Statement

For real numbers

x, y

and

z

it is known that

\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}

Find the value of the expression

x^2 + y^2+ z^2