The key result on quadratic functions as compositions is very useful here for visualizing how to solve an equation of the form $f(x)=Ax^2 + Bx +C= 0$ with $A \ne 0$ by considering the related equation $A(x−h)^2+k = 0$.

The mapping diagram for $f$ considered as a composition when $k<0$ helps visualize the two solutions, $x=h \pm \sqrt{-\frac kA}$

You can use this next dynamic example to solve quadratic equations like that in Example QEQ.0 visually with a mapping diagram of $f$.

A standard form of a quadratic equation with variable $x$ is an equation of the form $Ax^2 + Bx +C= 0$ with $A \ne 0$.

A beginning algebra course spends a considerable amount of time solving the standard quadratic equation without any reference to functions.

The relation between the coefficients $A,B,C$ and $h$ and $k$ is most easily seen by expanding:

$A(x−h)^2+k = Ax^2 -2Ahx +Ah^2 +k$.

So $B= -2Ah$ and $C=Ah^2+k$. This gives $h=-\frac B{2A}$ and $k=C-\frac {B^2}{4A}$.The solution of the conventional form of the quadratic equation is $x=h \pm \sqrt{-\frac kA} = -\frac B{2A} \pm \sqrt{-\frac {C-\frac {B^2}{4A}}A} =-\frac B{2A} \pm \sqrt{-\frac {4AC-B^2}{4A^2}}\\ x =-\frac B{2A} \pm \frac {\sqrt{B^2-4AC}}{2A} =\frac{-B \pm \sqrt{B^2-4AC}}{2A} $

A slightly more ambitious form of a quadratic equation with variable $x^2$ is an equation of the form $A_1x^2 + B_1x +C_1 = A_2x^2 +B_2x + C_2$ with $A_1 \ne A_2$. Algebraically this is solved by solving the related equation $Ax^2 + Bx+C= 0$ where $A = A_1 - A_2$, $B = B_1 - B_2$, and $C = C_1 - C_2$.

You can use this next dynamic example to solve quadratic equations like those in Examples QEQ.1 and QEQ.2 visually with a mapping diagram of $f$ and the parabolic curve in the graph of $q$.