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I have this equation: $x^2-4x+5 = 0$ . Its roots are imaginary $2 \pm i$ and I read in Algebra book/resource somewhere that to graph a quadratic equation with imaginary/complex roots, you need a complex-plane. But I can graph this (it is a parabola) with real numbers without any need for complex plane. What gap is there in my knowledge of algebra ?

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There is no need of the complex plane to graph a quadratic function. But you need it in order to see the roots of your specific quadric.

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