Problem

Determine the real intersections of \(y=x^{2}+2\) and \(y=x\). Form the quadratic by substitution, calculate its discriminant, classify the system, and relate the result to the displayed answer graph.

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set equal=x²+2=x; reduced=x²−x+2=0; discriminant=(-1)²+4·1·2=9>0; classification=two real intersections; graph meaning=line crosses parabola twice

set equal=x²+2=x; reduced=x²−x+2=0; discriminant=(-1)²−4·1·2=-7<0; classification=no real solution; graph meaning=line and parabola do not intersect in the real plane

set equal=x²+2=x; reduced=x²−x+2=0; discriminant=-7; claimed roots=x=(1±sqrt(7))/2; classification=two real intersections; graph meaning=line crosses parabola twice

comparison=x²+2≠x; reduced=x²−x+2≠0; test=the expressions differ for tested values; classification=no real solution; graph meaning=graphs do not meet

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Determine the real intersections of \(y=x^{2}+2\) and \(y=x\). Form the quadratic by substitution, calculate its discriminant, classify the system, and relate the result to the displayed answer graph.

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