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Saturday August 23rd 2014

Where Did the Quadratic Formula Come From?

Historic Guilford House (1892) at Case Western Reserve University in Cleveland. Photo courtesy of Standard Travel Photos click for larger image

Many of you probably recall the quadratic formula from high school algebra. You were given a quadratic equation of the form  a quadratic equation and had find the roots of the equation. If you recall, when a  quadratic equation is plotted, it usually takes the form of a parabola. And there are normally two places where the parabola crosses the x-axis, so there are normally two zeroes, or two roots.

One way to solve for the roots, or zeroes, was to factor the left side of the equation into the form (x+r1)(x+r2) = 0. Then you set each factor to zero to find the two roots: x+r1=0 or x+r2=0; then the roots are x=-r1 or x=-r2

Quadratic Formula
Often the equation was not so easy to factor. In that case, the quadratic formula was used to find the roots.

quadratic formula

We all remember this quadratic formula, some perhaps not so fondly, because you might end up with the square root of some monstrosity. (Much of this was mitigated by the development of electronic calculators. Those who had to do it with pencil and paper were not so lucky.)

But where did this formula come from? Who figured it out? Well you can learn the answer by watching these two video presentations from Khan Academy.

Completing the Square
The first video shows how to solve a quadratic equation by “completing the square”. The idea is to make use of the observation that (x+a)2 = x2 + 2x + a2 and then get the left side of our quadratic equation into that form. Watch the video.

Derivation of Quadratic Formula
The second video shows how the quadratic formula is derived by applying the idea of completing the square to the general quadratic equation a quadratic equation

You can learn more about algebra and many other academic subjects at Khan Academy

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