Visualizing the quadratic formula

追加: 2026-06-07

[00:00:00]Did you know you can actually visualize the quadratic formula? Yes, that thing you learned in high school. Today I'll show you that you can actually see the quadratic formula with your own eyes.

[00:00:08]See, in school you learned about straight lines and you learned that this formula, Y is equal to MX plus B, basically tells you the slope of the line M and the Y intercept B. Now you also learned how to graph straight lines on an XY plane. Then you learned about these things called parabolas. And a parabola is basically the trajectory of any projectile, for example, throwing a free throw And you also learned about the famous shape of a parabola where it's either opening up or it's closing down. And then you learned about the formula of the parabola, which is Y is equal to AX squared plus BX plus C. And here came the big question, where does the parabola intersect the X axis? Now out of the blue your teacher probably threw a formula at you, which is this X is equal negative B plus or minus the square root of B squared minus 4AC all over 2A. That's how I memorize it. And basically this formula tells you where the parabola crosses the X axis. But how do those red points relate to that formula? Can we actually visualize the solution? The answer is yes. I'm going to do some simplifications here, but if you understand this example, you can understand the general case. Starting with the most general parabola, let's set this A equal to 1 because it's not going to change any of the geometric intuition. So we're going to set it equal to 1. This C right here is merely a constant that shifts your parabola up or down. So for now we're going to set it equal to zero and just forget about it. This gives us the parabola Y is equal to X squared plus BX. Now if we solve the quadratic formula, we find that the solutions are of this form. Can we make sense of this geometrically? You see the parabola we just looked at had solutions X is equal to zero and X is equal to minus B. And if we plot the parabola, we can see that it crosses negative B and it crosses zero. But there's something very interesting about parabolas and that is that they are symmetric, which means there's always a line of symmetry that breaks the parabola in half, where the left is basically a mirror image of the right.

[00:02:07]Now, in our example, this line of symmetry was -b/2. Now, if we focus on the solution of the quadratic formula, notice what it's telling us. In red here, first it's telling you you should move towards the line of symmetry -b/2.

[00:02:22]Then in green, it's telling you move either to the right or to the left.

[00:02:26]That's the plus or minus. Okay, so I start at -b/2, and then I should move to the left and to the right, but by how much? Well, in blue tells you exactly how much you should move to the left and to the right in order to figure out the solutions. So, there you have it. In the simplest case, the quadratic formula tells you first hop to the line of symmetry, then jump either to the left or to the right by a fixed amount.

[00:02:50]That's where the parabola crosses the x-axis.