From a trinomial to the sign on the axis
The inequality ax2+bx+c≷0, a=0, can be solved in two consistent ways:
1) Geometrically: the graph y=ax2+bx+c is a parabola; find where it lies above or below the x-axis.
2) Algebraically: factor a(x−x1)(x−x2) when D>0 (or handle a double root when D=0, no real roots when D<0) and apply the sign-chart method to the factors.
Both approaches give the same set of x.
If a>0, branches open up — a "bowl"; if a<0 — down
Roots are x-intercepts (if any exist)
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