Quadratic Formula: |
This investigation is on the quadratic formula, which gives the solutions of a quadratic equation. In many secondary math classes, students are expected to memorize this formula and know how to use it. Unfortunately, they do not understand why this formula works or where it comes from. The first part of this investigation will explain the derivation of the quadratic formula. The second part will explore various concepts related to the quadratic formula and tie in college level concepts.
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Derivation of the Quadratic Formula
The quadratic formula gives us the values of our input, x, when our output, y, is zero in a quadratic equation. Therefore it tells us the x-intercepts (also known as solutions, roots, or zeroes) of a quadratic function. If we start with a quadratic equation in standard form, like the one on the right, where a, b, and c are numerical coefficients, we can derive the quadratic formula by assuming y=0 and solving for x.
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To solve for x, we need to use the method of completing the square. This method allows us to solve for x by creating a perfect square trinomial, which can easily be factored. A perfect square trinomial is an equation of the form a^2+2ab+b^2 which can be factored into (a+b)^2. This helps us solve for x because it will allow us to write all the terms associated with an x as one term. Once all the terms with an x are combined, we can solve for x by isolating the variable.
The best way to visualize what must be added to the equation to complete the square is by using geometry. We can represent all the terms with an x in them as shapes and arrange them in such a way that we can visually see what must be added to complete the square. This is shown below, along with the entire derivation process:
The best way to visualize what must be added to the equation to complete the square is by using geometry. We can represent all the terms with an x in them as shapes and arrange them in such a way that we can visually see what must be added to complete the square. This is shown below, along with the entire derivation process:
The quadratic formula is helpful because it allows for quick solutions to any quadratic equation. Since the completing the square method was used in a general case, instead of having to use this method every time, we can just use the quadratic formula to find the roots of the equation. All we would need to do is write the equation in standard form and identify what a, b, and c are.
Types of Solutions
Another useful tool that comes from the quadratic formula is using the discriminant (the expression under the square root in the formula) to determine what kind of solutions the equation will have. There are three possible kinds of solutions.
We can see examples of each of these graphically. All quadratic equations are parabolas. If there is one real solution, the vertex of the parabola will be on the x-axis. If there are two real solutions, the parabola will intersect the x-axis at two points. If there are two complex solutions, the parabola will not intersect the x-axis at all.
Parameters of Quadratic Equations
The quadratic formula gives us the solutions for x by using the parameters of the original quadratic equation (a, b, and c). Therefore, it is important to understand how each of these parameters influence what the graph of the function will look like. The following Desmos Activity allows you to change the values of a, b, and c and observe how each of them change the shape of the graph of the equation. Click on the picture below to access the graph.
The sign of a changes whether the parabola faces up or down. The value of a determines whether the parabola is narrow or wide.
The b is the slope of the tangent line that intersects the graph at the y-intercept. As a result, the b value affects the position of the vertex of the parabola. The c is the y-intercept. When x=0, the value of the function will be c. Thus b is the slope of the tangent line at the point (0, c). We also know that the height of the graph is determined in part by c. |
Comparing Quadratics with Complex Roots to Quadratics with Real Roots
When the roots of a quadratic equation are real, it is easy to see them on a graph, because they are the x-intercepts. When the roots are complex, it is harder to determine how they are represented in the graph.
To begin this process, I started by comparing my function with complex roots to a function with real roots. There are infinitely many quadratic functions that graph parabolas, so I chose to compare my graph of a quadratic with complex roots to the flipped quadratic. That is, if we draw a horizontal tangent line through the vertex of the quadratic function, then reflect our function about this line, we will get a flipped quadratic with real roots. You can access the next activity by clicking on the image to the right. |
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I started this exploration by graphing the quadratic equation f(x). I created sliders for a, b, and c. My first instinct was to graph g(x) to get the 'flipped' quadratic that I wanted to compare my original to. This gave me a new quadratic function, g(x), that was opened in the opposite direction, but it was only reflected about the vertex's tangent when b=0. Otherwise, the new function would not be reflected across the line I wanted.
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I realized that although the sign of the parameter a does determine which way the parabola opens, the parameter b also plays a role in both the horizontal and vertical positioning of the vertex of the parabola. Therefore, to get the reflection of my original quadratic, I needed to write it in vertex form so that changing the sign of a would in fact give me the reflection of my original quadratic.
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In the activity, I have graphed all three equations, f(x), g(x), and h(x). You can see that if you change the slider for b, f(x) and h(x) will always be the reflection of each other about their vertex, but g(x) is not. Therefore, I decided to compare f(x) and h(x) to see if I could find a relationship between the real roots of f(x) and the complex roots of h(x). (I also want to point out that if you use the slider for b and c, sometimes f(x) will have complex roots and h(x) will have real roots. I assumed h(x) had complex roots for my investigation.)
To see the relationship between the roots of f(x) and h(x), I first needed to find them using the quadratic equation. |
Graphical Representation of Complex Roots
In my comparison of a quadratic with complex roots with its 'flipped' partner, I realized that the roots of the two quadratics, one with real and one with complex roots, were almost identical. Graphing real roots of a quadratic equation is fairly straightforward, because the real roots are the x-intercepts of the parabola. Therefore, I decided to try and use the graph with the real roots to help me find the graphical representation of the complex roots of the original quadratic.
It is important to first understand how to graph complex numbers in the complex plane. This Geogebra activity (click on the image on the left) by 'sonom' will help review how this is done. The real part of a complex number determines the horizontal positioning along the x-axis and the complex part of the number (the part with i) determines the vertical positioning along the i-axis. Therefore, if each real number is a horizontal vector in the imaginary plane, then multiplying the real number by i will rotate the vector 90 degrees, in the counterclockwise direction.
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I started by graphing two quadratic functions. One with real roots, r(x), and one with complex roots, n(x). You can follow along with this part of my investigation by clicking the picture to the right. I then labeled the real roots on the graph of r(x). Even though I was working in the (x, y) plane, I graphed the complex roots so that the real part of the root was the x-coordinate and the complex part of the root was the y-coordinate. When comparing the two sets of roots, real and complex, I found the distance between the pairs of roots. The distance between the real roots of r(x) was equal to the distance of the complex roots of n(x). Therefore, I decided my next step was to find a way to write the quadratic r(x) in the complex plane and hopefully I would see the complex roots of n(x).
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Click the image above to access the graph! |
For the three dimensional graph, that includes the x, y, and imaginary axes, I modified the activity by Kevin Hopkins in Geogebra. His original activity can be found here. I first graphed r(x) and n(x) in the (x, y) plane, which is the 'ground' of the 3D graph. The axis labeled z is the imaginary axis in this case. I know that in the imaginary plane, multiplying by 'i' rotates a vector 90 degrees counterclockwise, so if I want r(x) to be in the complex plane, I want to rotate it by 90 degrees. In my graph, I showed a paraboloid that represents the rotation of r(x) by any degree. If we observe just the parabola that is on the (z, y) plane, we will see that the rotation of r(x) intersects the z-axis (which is the imaginary axis) at two points, which are complex roots. I know from my previous investigation that the distance between the real roots of r(x) and the imaginary roots of n(x) is the same. If I draw a circle that intersects the real roots of r(x) and goes around the paraboloid. I will find that the circle also intersects the complex roots. Since the distance between both sets of roots represent the diameter of the circle, I know that they are equal. Therefore, I can conclude that the intersection of the rotated image of r(x) intersects the imaginary axis at the complex roots of n(x)!
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Conclusion
Throughout my investigation, I derived the quadratic formula, explored the various types of solutions to quadratic functions, and then dove deeper into how complex solutions could be graphically represented.
The quadratic formula can be derived using completion of the square, which can easily be seen with geometry and literally making a square. Given the quadratic formula, the discriminant will determine whether the quadratic equation will have one real solution, two real solutions, or two complex solutions.
To graph the complex roots of a quadratic equation, the first step is to reflect the quadratic about the tangent line of its vertex. This will result in a new quadratic equation with real solutions. Then, after rotating the quadratic with real solutions by 90 degrees, so that it lies in the imaginary plane, the rotated quadratic will intersect the imaginary axis at the two complex roots of the original quadratic.
The quadratic formula can be derived using completion of the square, which can easily be seen with geometry and literally making a square. Given the quadratic formula, the discriminant will determine whether the quadratic equation will have one real solution, two real solutions, or two complex solutions.
To graph the complex roots of a quadratic equation, the first step is to reflect the quadratic about the tangent line of its vertex. This will result in a new quadratic equation with real solutions. Then, after rotating the quadratic with real solutions by 90 degrees, so that it lies in the imaginary plane, the rotated quadratic will intersect the imaginary axis at the two complex roots of the original quadratic.
In my personal investigation, I referenced the following article on graphical interpretations of complex roots. All of the math and explanations were my own work, but they were inspired by the explanations in this article.
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