A Geometric Interpretation of the Quadratic Formula
AA Geometric Interpretation of the Quadratic Formula
Chenguang Zhang ∗ January 5, 2020
Abstract
This article provides a simple geometric interpretation of the quadratic formula. The ge-ometry helps to demystify the formula’s complex appearance and casts it into a much simplerexistence, thus potentially benefits early algebra students.
This work came from a note I wrote down several years back. Motivated by the recent coverage ofPo-Shen Lo’s work [1], I decide it could worth sharing. It is my hope that this short article will addyet another pleasant aspect to the time-honored quadratic formula. To keep it short, the author willnot review the rich history behind the quadratic formula, and instead highly recommend readersto consult Lo’s article [1] and references therein.
The quadratic formula gives the roots to the algebraic equation ax + bx + c = 0. Without loss ofgenerality, we set a = 1 and assume b > c >
0. So we are left with x + bx + c = 0 (1)to solve. Let us first start with a simpler function (or curve) y = x + bx. (2)Algebraically, its roots are solutions to the equation y = x + bx = 0, which are clearly x = 0and x = − b . Geometrically, its roots are the intersection of the curve with the x -axis, which aremarked by the two red dots in Fig. 1.Before adding c back, we should realize two geometric properties of curve 2 (see Fig. 1).1. Both the curve and its two roots are symmetric about x = − b/ c back, we merely lift the curve vertically upward. Thus, the symmetryis respected. ∗ Department of Mathematics, Massachusetts Institute of Technology. Email: [email protected] . a r X i v : . [ m a t h . HO ] J a n c b/2 √ Δ /2 Figure 1: The curves y = x + bx (solid, associated with the light-shaded region) and y = x + bx + c (dashed, associated with the dark-shaded region). The green arrows indicate how the two curvesare related: by a vertical shift. 2. For easy reference, we refer to the region between the x -axis and the part of curve below itas the “valley”. They are shaded in Fig. 1. The “half-width” of this valley is b/
2, and the“depth” of this valley is b /
4. While the depth can be found by directly substituting x = − b/ by definition . Naturally, the depth of the valley (here the vertical shift) equals the squareof the half width of the valley (here the horizontal shift). Conversely, the half-width of thevalley equals the square root of the valley depth.Now we add c back. By item 1, adding c lifts y = x + bx vertically upward by an amount of c , so the valley is shallower and its new depth is b − c . By item 2, we know that the half widthof the new valley is (cid:113) b − c . Because the two new roots are still symmetric about x = − b/
2, wehave the roots − b ± (cid:114) b − c, (3)or in the more familiar form: − b ± √ b − c . (4) In this article, the author splits Eq. (1) into x + bx = 0 and + c to derive the geometric inter-pretation. He also tried other ways of splitting. For example, x + c and bx , or x and bx + c .They failed to work due to that the term bx or bx + c represents a slanted line, making it hardto proceed. It is likely that the split used by this article is the easiest one that yields to cleargeometric interpretations. An interesting observation is that the discriminant of the quadratic equation ∆ = b − c is nothingbut the full width of the valley squared . Certainly, there will be no roots when this value is negative.Yet another interpretation is that the depth of the valley is ∆ = b − c . When it is negative,the valley is non-existent, the curve is above the x -axis (by an amount of − ∆), and there are noroots.At this point, it is clear that a positive c always tries to eliminate the valley and roots. Whereas b , regardless of its sign, always creates two roots. In this sense, the two coefficients compete againsteach other. For the students to follow this article, it is vital that they first comprehend the geometry of quadraticcurves. The part likely needs the most explanation is item 2 of the previous section, which relatesthe horizontal and vertical changes of the curve, as one travels along it from its very bottom.As a final comment. Eq. (3) seems to have clearer meaning than Eq. (4). In the author’sopinion, there are clear geometric interpretations of each value and operator in Eq. (3); a heavily3igure 2: Eq. (3), with each term annotated by its geometric meaning .annotated version of which is shown in Fig. 2. Much of these interpretations are lost in Eq. (4),leaving a lifeless instruction waiting to be executed mechanically.
References [1] Loh, Po-Shen. (2019).
A Simple Proof of the Quadratic Formula . arXiv preprintarXiv:1910.06709arXiv preprintarXiv:1910.06709