Quadratics and Complex Summary

Algebra 2 · Axiom Academy

SUMMARY Quadratics and Complex Numbers Let's review how quadratic functions model physical phenomena and how complex numbers complete our understanding of polynomial solutions. Complex numbers give every quadratic equation exactly two solutions in , counting multiplicity — a "no real roots" verdict just means the roots are complex. Standard, vertex, and factored form are three names for one parabola — pick the form that already shows the feature you need: intercept, vertex, or roots. The discriminant predicts the type of solution before you solve anything: two real, one repeated real, or a complex-conjugate pair. Parabolas also have a purely geometric definition — the set of points equidistant from a fixed focus and a fixed directrix. Complex numbers and polynomial solutions aren't a detour — they're the foundation for calculus, differential equations, and beyond. Looking ahead: the Fundamental Theorem of Algebra says a degree- n polynomial has exactly n roots in , counted with multiplicity — quadratics are just the n=2 case you've mastered here. Core Concept Three Forms of Quadratics Standard form reads off the y -intercept (0,c) with no work. Vertex form, y=a(x-h)^2+k , reads off the vertex (h,k) directly — from standard form, (mind the minus — dropping it is the classic error) and k=f(h) . Factored form, y=a(x-r_1)(x-r_2) , reads off the roots — but it only exists over when ; a parabola that never crosses the x -axis (like y=x^2+1 ) has no real factorization.

This is the written version of the interactive lesson above. See the full Algebra 2 course.