Problem

Analyze the real intersections of \(y=x^{2}+5\) and \(y=2\). Set the expressions equal, reduce the equation, justify the real-solution classification, and connect it to the displayed answer graph.

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What this problem is really about

A real intersection would require the two expressions for y to be equal. We’ll simplify that equality, use the fact that a real square cannot be negative to classify the system, and confirm the conclusion by comparing the horizontal line with the parabola’s minimum height.

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set equal=x²+5=2; reduced=x²=3; roots=x=±sqrt(3); classification=two real intersections; graph meaning=line y=2 crosses parabola twice

set equal=x²+5=2; reduced=x²=-3; claimed roots=x=±sqrt(3); classification=two real intersections; graph meaning=line y=2 crosses parabola twice

comparison=x²+5≠2; reduced=x²≠-3; test=most real x satisfy the inequality; classification=no real solution; graph meaning=graphs do not meet

set equal=x²+5=2; reduced=x²=-3; real-solution test=a real square cannot be negative; classification=no real solution; graph meaning=line y=2 lies below parabola whose minimum is5

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Analyze the real intersections of \(y=x^{2}+5\) and \(y=2\). Set the expressions equal, reduce the equation, justify the real-solution classification, and connect it to the displayed answer graph.

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