The Quadratic Formula

The quadratic formula is one of the most important expressions taught to math students, for reasons that extend much farther than solving quadratic equations. As it turns out, as also discussed in Rational Algebraic Expressions, second degree polynomials, or polynomials that have an x-squared in them, appear throughout science and Nature as a result of solving differential equations using a special technique called a transform. This fact is actually responsible for the appearance of oscillating systems in Nature, such as water waves, sound waves and light, as the quadratic formula is an excellent opportunity to introduce the concept of imaginary and complex numbers.

So let us revisit what the Quadratic Formula, shown above, actually is: it is a formula that helps us to factor a second-degree polynomial (a polynomial with an x-squared in it) when the factors are not readily apparent (which they very rarely are in real world situations). If you remember, when we are faced with an algebraic equation that looks like this,

x² + 5x + 6 = 0 (Equation 1)

we can sometimes solve this equation by factoring the second degree polynomial on the left hand side, and then setting each of these factors equal to zero. So, when we factor the polynomial above, we can rewrite Equation (1) as

(x+3)(x+2) = 0

And it turns out that this is only true when x+3 = 0, or x+2 = 0.

The reason this works is that if any product of two numbers is equal to zero, then either of those numbers, or both of those numbers must be zero, in order for the product to be zero. Any non-zero number times another non-zero number can never be zero, regardless of how small or negative either of them are.

And so if (x + 3) times (x+2) is equal to zero (which is the same thing as x + 5 x + 6 = 0), then either x +3 = 0 or x + 2 = 0. There can be no other way. And so, in this case, x = -3, or x = -2, which are called the roots of the polynomial. To see that this works, we can then plug in these values of x into Equation 1 and we will see that these values of x do indeed solve the original equation.


But how can we solve this problem if we cannot factor our polynomial into two factors? This is when the quadratic formula is used. We won't go through the derivation of the QF, which utilizes a principle called completing the square, but it is good to convince yourself that it works by looking at derivations like this, this and this. Once you convince yourself that the quadratic formula works, we can begin to understand how it works and what it can tell us about an equation we are trying to solve.

GRAPHING SECOND DEGREE POLYNOMIALS

Just as understanding the equation of a line, y = mx + b could enable us to graph a line on a coordinate axis using the slope 'm' and the y-intercept 'b' (more here), similarly, we can look at the general equation of a second degree polynomial, ax^2 + bx + c, and figure out what this equation would look like if we graphed it. By doing so, we can understand a lot about the particular equation we are solving.

If you recall from the posts regarding the equation of a line, it is helpful to understand functions by treating them in their most general form. In the case of a line, the function we considered was

y(x) = ax + b

and we learned that for different values of a and b, the graph of the function y(x) would look different. Because the highest power of x in the function y(x) is one, the equation of a line is an example of a FIRST DEGREE POLYNOMIAL. Because there are two parameters that we can control to make our line look different (a and b), we say that the equation of a line has two DEGREES OF FREEDOM. Any line in two dimensions can be graphed using the equation above by controlling the values of a and b.

Similarly, we can arrive at any parabola seen in Figure ## by changing the values of a, b and c in equation (2). Just as there was a rationale to how different values of a and b would affect the type of line we would get (the amount of slope, whether the slope was up or down, and where the line would intercept the y-axis), so too there is a systematic rationale to understanding how to make a particular parabola by changing the values of a, b and c.

One of the big leaps that happens when considering the parameters of a second-degree polynomial (a, b and c), compared to a first-degree polynomial (m and b), is that the shape of our graph -- in this case a parabola -- depends not just directly on the parameters themselves, but rather on the relationships of the parameters. In the case of a parabola, the value of 'a' does tell us the direction our parabola will open (up or down) and the value of 'c' will tell us the y-intercept of the parabola (the place where it will cross the y-axis). But these characteristics themselves do not provide enough information to know what the entire parabola will look like.

The first thing to notice about all of the parabolas in Figure 2 is that we can draw a vertical line through each of them about which the graph is symmetric -- if we fold the parabola along this line, it will fold back onto itself. This line is therefore called the axis of symmetry, and is arrived at by the equation:

x = -b/2a

It is important to notice that since this equation is x equalling a constant value, it represents the equation of a vertical line. Once we draw this line, we can begin to see where our parabola will be located on the coordinate axis, even though we still don't know exactly what it will look like and where it will be. What we can see, however, for each of the parabolas in Figure 2 is that the value that the parabola takes at the axis of symmetry is going to be the biggest or smallest value that the parabola ever takes (depending on if it is going up or down). As you will learn, a parabola can only change directions once, and does so at the axis of symmetry. Since the axis of symmetry only gives us an x-coordinate, we need to plug in that x-coordinate to our original quadratic equation to determine what the value of y is at the axis of symmetry. The x-y coordinates of the parabola at that point is called the VERTEX. Once we find the vertex, we can position our parabola vertically along the axis of symmetry.

THE ROOTS

Once we know where the vertex is, and if our parabola opens up or down, the next step is to figure out how wide or narrow our parabola is going to be. There are a few ways to do this, but the first way we will consider is to figure out the places where our parabola crosses the x-axis, which are referred to as the roots of our polynomial.

As mentioned above, in cases where we cannot factor our polynomial we will use the quadratic formula to determine the roots. If you notice from the QF written above, there are actually two solutions for the roots due to the +/- that appears before the square root. So in fact, the roots of a polynomial ax^2 + bx + c are given by


Though this may look rather complicated, we can break it down to make it much easier to grasp.

If you notice, the roots each have the term -b/2a in them, and then either add or subtract the same quantity from -b/2a: . So, the roots will be equally spaced around the value -b/2a, which they should be because we said that parabolas are symmetric around the line x = -b/2a.