How does the inflection point of quadratic equations relate to the quadratic formula?

The quadratic formula is one of the most useful formulas that are taught since high school, and for good reason. This formula allows to find the values of x for which a quadratic equation equals 0.

The quadratic formula

However, where does this formula come from and how does it relate to the inflection point of a quadratic equation? To determine such questions there are two important things to keep in mind for quadratic equations:

1. Every quadratic equation has one and only one inflection point
2. Every quadratic equation is symmetric around that inflection point

What is the inflection point of a quadratic equation?

The inflection point of a quadratic equation is the point x in which the quadratic equations reaches either its maximum or minimum value y.

Since the inflection point is the point in which a quadratic equation is symmetric, it is possible to obtain its value from two points of x that are equidistant around the inflection point, and then adding them and divide them by 2 to get the middle value, which is the inflection point. For example, by equating a quadratic equation to c:

Equation I

Of course, the above is solved for x, for which x₁ = 0 and x₂ = -b / a are the solutions:

• x₁ = 0

Equation I.I

• x₂ =-b / a

Equation I.II

Now the the inflection point, is simply obtained as follows:

Quadratic equation with its inflection point marked as a dash line. Graphed in Desmos.

How does the inflection point relates to the quadratic formula?

The quadratic formula gives the values of x for which the quadratic equation equals 0. Such values are equidistant around the inflection point x = -b / 2a, so it is possible to add or subtract some value ρ to the inflection point that results in the values of x for which the quadratic equation equals 0:

Equation II

But what is the value of ρ? To answer that question we have to examine the elements of a quadratic equation, usually defined by:

The above can be rearranged as:

Such that:

With the quadratic equation in this form, the values of x, when the equation is equal to 0, are easily found:

With this new information, it is now possible to express the equation II in the following terms:

Equation III

By multiplying both expressions in equation III:

Equation IV

Solving the above for ρ:

Equation V

Finally, retaking the equation II

it is possible to solve the values of x by replacing ρ with the expression obtained in equation V: