Quadratics

7.2 Properties of the Quadratic Formula

Some curricula feel it is important to notice that the formula

\(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

represents two symmetrical values about the middle point \(x=-\frac{b}{2a}\).

From part 4 of this course we know that the vertex of the parabola lies halfway between any two symmetric points. Our technique of simply looking for interesting \(x\)-values makes the location of the vertex clear. One need not know this formula.

(But if you do want it … just write \(y=ax^{2}+bx+c\) as \(y=x(ax+b)+c\). This shows that inputs \(x=0\) and \(x=-\frac{b}{a}\) give symmetrical outputs. The vertex is thus halfway between these values … at \(x=-\frac{b}{2a}\)!)

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