Applications of Solving Factorable Quadratic Equations

When a polynomial is set equal to a value (whether an integer or another polynomial), the result is an equation. An equation that can be written in the form [latex]ax^{2}+bx+c=0[/latex] is called a quadratic equation. You can solve some quadratic equations using the rules of algebra, applying factoring techniques, and by using the Principle of Zero Products.

There are many applications for quadratic equations. Don’t forget that to use the Principle of Zero Products to solve a quadratic equation, you need to make sure that the equation is equal to zero. For example, [latex]12^{2}+11x+2=7[/latex] must first be changed to [latex]12x^{2}+11x-5=0[/latex] by subtracting 7 from both sides.

Area Problems

In our first example, we use some basic geometry as we search for the dimensions of rectangle.

In the example in the following video, we present another area application of factoring trinomials.

Consecutive Integers

Our next example is deceptively similar to the rectangular garden (see if you can recognize this). First, we must recall what is meant by “consecutive integers.” Two consecutive integers would be something like 7 and 8 or -53 and -52. We can also explore consecutive odd integers (such as 15 and 17) or consecutive even integers (like 84 and 86).

Graphing Basic Quadratic Functions

In Modules 5 and 6 we learned about polynomials. An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as [latex]{x}^{2}-3x-4=0[/latex] and [latex]{x}^{2}-16=0[/latex] are in the “family” of quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Just like the linear functions we’ve learned about previously, quadratic functions can also be graphed. It is helpful to have an idea about what the shape should be so you can be sure that you have chosen enough points to plot as a guide. Let us start with the most basic quadratic function, [latex]f(x)=x^{2}[/latex].

Graph [latex]f(x)=x^{2}[/latex].
We’ll start with a table of values. Then think of each row of the table as an ordered pair.

Now we’ll plot the points [latex](-2,4), (-1,1), (0,0), (1,1), (2,4)[/latex]

Notice that the quadratic family of functions is not drawn using straight lines. Since the points are not on a line, you cannot use a straight edge. Connect the points as best you can using a smooth curve (not a series of straight lines). You may want to find and plot additional points (such as the ones in blue below). Placing arrows on the tips of the lines implies that they continue in that direction forever.

Notice that the shape is similar to the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The lowest point on this graph is called the vertex. The vertical line that goes through the vertex is called the line of symmetry. In this case, that line is the y-axis, which is the line [latex]x=0[/latex].

In the following video, we show an example of graphing another quadratic function, [latex]f(x)=\frac{1}{2}x^2[/latex], using a table of values.

The equations for quadratic functions can be written in the form [latex]f(x)=ax^{2}+bx+c[/latex] (where [latex] a\ne 0[/latex]). In the two basic quadratic functions we graphed above, notice there was only the [latex]ax^2[/latex] term and that both [latex]b=0[/latex], and [latex]c=0[/latex] for those functions.

Although there are many forms that quadratic functions can be written in, this Elementary Algebra course will focus primarily on graphing basic quadratic functions of the form [latex]f(x)=ax^{2}+c[/latex].

Quadratic functions of the form [latex]f(x)=ax^{2}+c[/latex] will always be centered around the y-axis which is the line [latex]x=0[/latex]. 

We have shared two examples above of graphing a basic quadratic function of the form [latex]f(x)=ax^{2}+c[/latex], where [latex]c=0[/latex], so let’s now explore how to graph a quadratic function of the form [latex]f(x)=ax^{2}+c[/latex] where [latex]c\ne 0[/latex] .

Graph [latex]f(x)=x^{2}-4[/latex].
Similar to the problems above, lets start with a table of values.

We’ll now plot the points [latex](-2,0), (-1,-3), (0,-4), (1,-3), (2,0)[/latex] on the graph.

If we connect all the points with a smooth curve we will have the graph of our function [latex]f(x)=x^{2}-4[/latex].

Let’s have you now try one on your own:

Example 3

Graph the function: [latex]f(x)=x^2-16[/latex]

Show Answer

Notice our function [latex]f(x)=x^2-16[/latex] is of the form [latex]f(x)=ax^{2}+c[/latex]. This means it will be centered around [latex]x=0[/latex].

We can start our table with any values we want, but will want to make sure to choose values on both sides of 0, as well as the value of [latex]x=0[/latex].

[latex]x[/latex]	[latex]f(x)=x^2-16[/latex]	[latex]f(x)[/latex]
4	[latex](4)^2-16=0[/latex]	0
-4	[latex](-4)^2-16=0[/latex]	0
0	[latex](0)^2-16=-16[/latex]	-16
2	[latex](2)^2-16=-12[/latex]	-12
-2	[latex](-2)^2-16=-12[/latex]	-12

Plotting the ordered pairs from the table above gives us:

Connecting the points with a smooth curve gives our final graph:

 

Lets try a few more examples.

Example 4

Graph the function: [latex]f(x)=x^2+5[/latex]

Show Answer

Notice our function [latex]f(x)=x^2+5[/latex] is of the form [latex]f(x)=ax^{2}+c[/latex]. This means it will be centered around [latex]x=0[/latex].

Create a table of values

[latex]x[/latex]	[latex]f(x)=x^2+5[/latex]	[latex]f(x)[/latex]
-2	[latex](-2)^2+5=9[/latex]	9
-1	[latex](-1)^2+5=6[/latex]	6
0	[latex](0)^2+5=5[/latex]	5
1	[latex](1)^2+5=6[/latex]	6
2	[latex](2)^2+5=9[/latex]	9

Plot the points (-2,9), (-1,6), (0,5), (1,6) and (2,9) and connect them with a smooth curve.

Extension: 

Notice some quadratic functions never cross the x-axis. The function [latex]f(x)=x^2 +5[/latex] is one example. If we try to solve for the x-intercepts in this function by setting the function equal to 0, we would have the following:

[latex]x^2 +5=0[/latex]

[latex]x^2 =-5[/latex]

[latex]\sqrt{x^2}=±\sqrt{-5}[/latex]

If we try and find the square root of -5 in our calculator we will find that it gives us an error message. There are no real number answers to the square root of a negative value. Thus this function does not have any x-intercepts on our graph.

As we can see with the graph we made above, the parabola falls above the x-axis.

 

In the next example we will explore how parabolas don’t always open upwards.

Example 5

Graph the function: [latex]f(x)=-3x^2[/latex]

Show Answer

Notice our function [latex]f(x)=-3x^2[/latex], is of the form [latex]f(x)=ax^{2}+c[/latex], where [latex] a=-3[/latex] and [latex] c=0[/latex]. We will start to graph this by choosing values surrounding 0.

*It is important that you understand how to evaluate a function like this correctly. For this reason, we will show the steps for the input value of -2 in detail:

[latex]f(-2)= -3(-2)^2[/latex]

We must make sure we follow the correct “Order of Operations” here. This means that before we multiply the [latex]-3[/latex] with anything, we must first raise the  [latex]-2[/latex] to the exponent of 2. When we raise [latex](-2)^2[/latex], we are multiplying [latex](-2)\cdot(-2)[/latex], which is equal to [latex]4[/latex].

[latex]f(-2)= -3(4)[/latex]

We can now multiply the -3 with the 4 to find the final output.

[latex]f(-2)= -12[/latex]

This process will be repeated with all the x values input in the table of values below:

[latex]x[/latex]	[latex]f(x)=-3x^2[/latex]	[latex]f(x)[/latex]
-2	[latex]-3(-2)^2=(-3)(4)=-12[/latex]	-12
-1	[latex]-3(-1)^2=(-3)(1)=-3[/latex]	-3
0	[latex]-3(0)^2=(-3)(0)=0[/latex]	0
1	[latex]-3(1)^2=(-3)(1)=-3[/latex]	-3
2	[latex]-3(2)^2=(-3)(4)=-12[/latex]	-12

Plotting the ordered pairs from the table above gives us:

Connecting the points with a smooth curve gives our final graph below. Notice that this is an upside down U-shape this time.

As we saw with the last example, with quadratic functions of the form [latex]f(x)=ax^{2}+c[/latex], changing the value of [latex]a[/latex] can change the width of the parabola and whether it opens up ([latex]a>0[/latex]) or down ([latex]a<0[/latex]). If [latex]a[/latex] is positive, the vertex is the lowest point, and the parabola opens up. If [latex]a[/latex] is negative, the vertex is the highest point, and the parabola opens down. Whereas Examples 3 and 4 showed that the value of [latex]c[/latex] moves the parabola up or down.

The next example reveals what can cause a parabola to move right or left.

Example 6

Graph the function: [latex]f(x)=(x-4)^2[/latex]

Show Answer

If we choose the same values as the previous examples, we will obtain some relatively large results and also will fail to see both sides of the parabola. The key is to focus on the value of [latex]x[/latex] that causes the expression inside the parentheses to equal to zero.  In this case, that would be [latex]x=4[/latex].  So, as we select values to plug in, we will choose values centered around [latex]4[/latex].

Create a table of values

[latex]x[/latex]	[latex]f(x)=(x-4)^2[/latex]	[latex]f(x)[/latex]
2	[latex](2-4)^2=(-2)^2=4[/latex]	4
3	[latex](3-4)^2=(-1)^2=1[/latex]	1
4	[latex](4-4)^2=0^2=0[/latex]	0
5	[latex](5-4)^2=1^2=1[/latex]	1
6	[latex](6-4)^2=2^2=4[/latex]	4

Plot the points (2,4), (3,1), (4,0), (5,1) and (6,2) and connect them with a smooth curve.

 

The key to the last example is we no longer focused on choosing values around [latex]x=0[/latex].  Instead, we considered what would cause the quantity inside the parentheses to equal to zero, which was [latex]x=4[/latex], plugging in this number and values on either side of it.  If the function had been [latex]f(x)=(x+4)^2[/latex], the primary [latex]x[/latex]-value of interest would have been [latex]x=-4[/latex].  We can describe this using the form given below.

Not all parabolas fall into the two forms given here.  While you will explore these in more detail in future math classes, we end with a preview, including an application of the factoring skills we have learned.

Graphing Quadratics Summary

Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for x, finding the corresponding y values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.

The shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[/latex], where [latex]a, b[/latex] and [latex]c[/latex] are real numbers and [latex]a\ne0[/latex]. The value of [latex]a[/latex] determines the width and the direction of the parabola, while the vertex depends on the values of [latex]a, b[/latex] and [latex]c[/latex].