Posted October 2, 2015 By Presh Talwalkar. Read about me , or email me .

Quadratic equations are very important in mathematics, with applications ranging from sending a rocket into space to calculating the best price to maximize profits.

The quadratic formula gives the solution to any quadratic equation, but it is also confusing.

In my new video, I offer a geometric interpretation that shows what the formula is about. While this is almost never taught in schools, this is how most ancient civilizations actually solved quadratics (with the exception they avoided negative numbers and negative areas).

I hope you enjoy the video.

The quadratic formula – an intuitive explanation

If you can’t watch there is a text summary below with screen captures from the video.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Solving quadratics

Can you solve x2 + 2x – 15 = 0?

There are 3 common ways to solve the problem. You can graph the equation, try to factor it, or use the general quadratic formula.

Graphing is a numerical approach, and factoring is a quick analytic method. But when those don’t provide a quick and exact answer, the quadratic formula is an algorithm that provides the answer.

The problem is the equation is confusing. Where does it come from, and why does it work?

There is a geometric explanation for it.

Solving visually

First I will add 15 to both sides so we have the equation Can you solve x2 + 2x = 15.

Each term can be represented by a rectangle whose area is equal to that term.

For example, x2 is represented by a square with a side length x, so it has an area of x2.

Similarly 2x is represented by a rectangle with one side x and another side 2, so it has an area of 2x.

We don’t need to specify side lengths for 15–we will just draw a rectangle with an area of 15.

So we’re looking for a value of x that makes the sum of the areas on the left (x2 + 2x) equal to the area on the right (15).

How can we do that? There’s a neat trick!

Completing the square

We’ll divide the rectangle 2x into half so it is two rectangles of area x with sides x and 1.

Now we’ll add those rectangles to the square.

We almost have a square with a side x + 1. We can “complete the square” by adding an orange square.

The orange square has two sides of length 1, so its area is 1. Also, in order to balance the areas, we have to add the same area to the right hand side shape.

Algebraically we need to add 1 to both sides of the equation.

The right hand side simplifies to 16.

We now have a square with side x + 1 that needs to have an area of 16. How can we do that?

Easy! We know a 4×4 square has an area of 16, so we can set the side length x + 1 equal to 4.

There is another solution too. A square with side lengths -4 x -4 also has an area of 16 (since -4 squared is +16). So we get another solution by setting the side length equal to -4.

Now we’ve solved x = 3 and x = -5.

These are the same solutions we found by graphing the equation and factoring it. But now we understand why this is the solution: it is the value that makes the areas of the shapes equal to each other.

General proof

Now we can generalize the method to solve any quadratic.

Suppose we have ax2 + bx + c = 0. How do we solve this?

Let’s re-arrange the equation a bit. Let’s subtract c from both sides to get ax2 + bx + c = 0.

Now let’s divide by a, which gives x2 + (b/a)x = –c/a.

This form is similar to the problem just solved. Each term can be represented by a rectangle whose area is equal to that term.

Now we do the same trick of completing the square.

Completing the square (general)

We’ll divide the rectangle (b/a)x into half so it is two rectangles of area x with sides x and b/(2a).

Now we’ll add those rectangles to the square.

We almost have a square with a side x + b/(2a). We can “complete the square” by adding an orange square as follows.

The orange square has two sides of length b/(2a), so its area is b2/(4a2).

Also, in order to balance the areas, we have to add the same area to the right hand side shape.

Algebraically we need to add b2/(4a2) to both sides of the equation.

The right hand side simplifies to (b2 – 4ac)/(4a2).

We now have a square with side x + b/(2a) that needs to have an area of (b2 – 4ac)/(4a2). How can we do that?

As before, we set the side length equal to plus or minus the area of the shape on the right hand side. This is a more complicated expression, but ultimately it simplifies to the quadratic formula.

And that’s it! We’ve solved that x = (-b ± √(b2 – 4ac))/(2a).

The quadratic formula is one of the most important results in algebra. But it also can be understood from a geometric perspective that comes from setting the areas of shapes equal to each other.

You can read more about the history of the quadratic formula and its geometric roots from Cornell Mathematics professor David W. Henderson: Geometric Solutions of Quadratic and Cubic Equations.

While researching this post, I came across a website that offers a visual interpretation of the quadratic formula that helped me design the graphics: the quadratic formula – But Why? Intuitive Mathematics.

And do check out my video which has some animations of the above derivation.

The quadratic formula – an intuitive explanation

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