Video Transcript
In this video, we will learn how to
solve quadratic equations using function graphs.
Let us recall the definition of a
quadratic equation. A quadratic equation is an equation
that can be written in standard form as 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero,
where 𝑥 is the variable, 𝑎, 𝑏, and 𝑐 are constants, and 𝑎 is not equal to
zero. We note that it is always possible
to rearrange a quadratic equation to be equal to zero, as shown here, by taking all
of the variables and the constant to one side of the equation.
Now, recall that when we solve a
quadratic equation, we find the values of 𝑥 for which the equation is
satisfied. One of the ways we can solve a
quadratic equation is by factoring. This means we rearrange the
quadratic equation into the factored form 𝑎 multiplied by 𝑥 minus 𝑝 multiplied by
𝑥 minus 𝑞 equals zero. From here, it is possible to deduce
that both 𝑥 equals 𝑝 and 𝑥 equals 𝑞 satisfy the equation and that they are,
therefore, solutions of the equation.
In this video, we will see how we
can also use a graphical method to solve a quadratic equation. To plot a quadratic equation, we
rewrite it as a function: 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. In other words, we replace zero
with the variable 𝑦. Note that we will often denote the
left-hand side of the function by 𝑓 of 𝑥 as shown. Writing our equation as a function
allows us to show graphically how 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 changes for
different values of 𝑥.
Suppose we then want to solve the
quadratic equation using this graph. Since the quadratic equation is
solved when it is equal to zero, we set 𝑦 equal to zero in the function and find
the values of 𝑥 for which the equation is satisfied. Thus, the solutions of the equation
are the 𝑥-values for which the function is zero, which we refer to as the roots of
the function. On a graph, these values are the
𝑥-coordinates of the points where the 𝑦-value is zero, which corresponds to the
points at which the graph crosses the 𝑥-axis.
Now, graphs of quadratic functions
have distinctive properties which can be used to help us identify points of interest
about an equation. Whether inspecting the graph of a
quadratic function or using an equation to draw the graph, the following points are
important to remember. The graph of quadratic functions of
the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 have distinctive parabola
shapes. They have a minimum vertex with the
curve opening upward when the value of 𝑎 is greater than zero, as shown in the left
graph. They will have a maximum vertex
with the curve opening downward when 𝑎 is less than zero, as shown in the right
graph.
Note that the value of 𝑎 cannot be
equal to zero, as this would mean that there is no 𝑥 squared term. And as such, the corresponding
equation would not be quadratic. A quadratic function can also be
arranged into the vertex form as 𝑦 is equal to 𝑎 multiplied by 𝑥 minus ℎ all
squared plus 𝑘, where ℎ, 𝑘 are the coordinates of the vertex of the parabola,
i.e., the turning point.
The graph of a quadratic function
is symmetrical about the vertical line 𝑥 equals ℎ. The 𝑦-intercept of the function 𝑦
equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 will always be at zero, 𝑐. The 𝑥-intercepts, where the curve
crosses the 𝑥-axis, will be the points at which 𝑦 equal zero. The 𝑥-coordinates of these points
are the roots of the function and correspond to the solutions of the original
quadratic equation. In a graph, we can identify these
points via inspection.
It is useful to remember that a
quadratic equation will have up to two real solutions. If an equation has two solutions,
the corresponding function will have a graph that crosses the 𝑥-axis twice. An equation with a repeated
solution will lead to a graph that has a vertex on the 𝑥-axis. Finally, an equation having no
solution will mean that the graph is entirely above or below the 𝑥-axis. In the graphs shown, the first
function has two real roots, the middle function has one real root where the graph
touches the 𝑥-axis, and the last function has no real roots.
Let us now look at an example where
we can apply these properties to find the solution to a quadratic equation using a
graph.
The diagram shows the graph of
𝑦 equals 𝑓 of 𝑥. What is the solution set of the
equation 𝑓 of 𝑥 equals zero?
Let us recall that the
coordinates of any point on the graph of a function are given by 𝑥, 𝑦. We are being asked to find the
solution set of the equation 𝑓 of 𝑥 equals zero, which is the set of values of
𝑥 for which the 𝑦-value equals zero. On this graph, this corresponds
to the points at which the curve crosses the 𝑥-axis, since 𝑦 equals zero at
these points. By inspecting the curve, we can
see that it crosses the 𝑥-axis at two points: when 𝑥 equals negative two and
when 𝑥 equals two. Therefore, the solution set is
negative two, two.
In this example, we saw that since
the curve crossed the 𝑥-axis twice, the equation had two solutions. Let us consider an example where
that is not necessarily the case.
The graph shows the function 𝑓
of 𝑥 equals 𝑥 squared minus two 𝑥 plus three. What is the solution set of 𝑓
of 𝑥 equals zero?
Here, we have been given the
explicit function that describes 𝑓 of 𝑥, but since we have the graph, we can
simply solve the equation graphically without factoring or using the quadratic
formula. Recall that the solution set of
𝑓 of 𝑥 equals zero can be found by identifying the points 𝑥, 𝑦 on the graph
where 𝑦 is equal to zero, which is where the curve intersects the 𝑥-axis. In this case, however, the
curve is entirely above the 𝑥-axis. For this reason, there are no
points at which 𝑦 equals zero. Hence, there are no real values
of 𝑥 that solve the equation. So the solution set is the
empty set, denoted as shown.
In our next example, we will
consider how to factor an equation and use the answer to figure out what the graph
must look like.
Solve 𝑥 squared minus 𝑥 minus
six equals zero by factoring, and hence determine which of the following figures
would be a sketch of 𝑦 equals 𝑥 squared minus 𝑥 minus six. Is it (A), (B), (C), (D), or
(E)?
As we have been asked to solve
𝑥 squared minus 𝑥 minus six equal zero by factoring, let us first remember how
to do this. We want to factor the equation
using the two unknown values 𝑝 and 𝑞 as shown. Matching the coefficients, we
can see that this requires 𝑝𝑞 to equal negative six and 𝑝 plus 𝑞 to equal
negative one. Since the product of 𝑝 and 𝑞
is negative, this means one of 𝑝 and 𝑞 is negative and the other is
positive. Let us assume 𝑝 is
negative. Thus, let us consider the four
possible pairs of 𝑝 and 𝑞 that multiply to make negative six. Out of these options, only 𝑝
equals negative three and 𝑞 equals two gives us 𝑝 plus 𝑞 equals negative
one. Thus, the correct factoring is
𝑥 minus three multiplied by 𝑥 plus two.
Having factored the equation,
we can solve it by setting it equal to zero and finding the values of 𝑥 that
satisfy the equation. This is satisfied when 𝑥 minus
three equals zero or 𝑥 plus two equals zero, which leads to 𝑥 equals three or
𝑥 equals negative two.
We now need to determine which
of the given figures is a sketch of 𝑦 equals 𝑥 squared minus 𝑥 minus six. Recall that the roots of a
function tell us what values of 𝑥 give us 𝑦 equals zero. This means we know at what
points the curve will cross the 𝑥-axis: 𝑥 equals three and 𝑥 equals negative
two. Considering the five graphs,
only one crosses the 𝑥-axis at these points: option (E).
Additionally, let us note other
aspects of the graph that we can use to verify the correct answer. We note that the 𝑦-intercept
of the graph is negative six. Recall that for a quadratic
function of the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, the 𝑦-intercept
is 𝑐. Since the function is 𝑦 equals
𝑥 squared minus 𝑥 minus six, this gives us 𝑐 is equal to negative six, which
is indeed correct. Furthermore, we can see that as
𝑎 is equal to one, the graph must open upward, which is exactly what happens in
graph (E). We can therefore conclude that
the correct answer is option (E).
In some questions, we will be given
a quadratic equation that has to be rearranged before we can solve it
graphically. We will now look at an example of
this type.
Find the solution set of the
equation 𝑥 squared is equal to three 𝑥 plus 10.
As we have been asked to find
the solution set of the equation 𝑥 squared is equal to three 𝑥 plus 10, we
must first rearrange it into the form 𝑓 of 𝑥 is equal to zero. Note that we can do this by
subtracting three 𝑥 and 10 from both sides to get 𝑥 squared minus three 𝑥
minus 10 equals zero. Recall that the solutions of
the equation 𝑓 of 𝑥 equals zero are the 𝑥-values of the points where the
graph of the function crosses the 𝑥-axis. We can find these values by
sketching the graph of the function 𝑓 of 𝑥 equals 𝑥 squared minus three 𝑥
minus 10 and finding where it crosses the 𝑥-axis.
We can draw the graph by
setting up a table of values and calculating 𝑓 of 𝑥 for the selected
𝑥-values. Taking values of 𝑥 from
negative three to six, we have the given values of 𝑓 of 𝑥. We can then plot these on the
𝑥𝑦-plane and join them with a smooth curve. It is easily seen that the
curve crosses the 𝑥-axis at 𝑥 equals five and 𝑥 equals negative two. Note that we could have read
off these values directly from our table, which showed that 𝑓 of 𝑥 equals zero
for these two values of 𝑥. We conclude that the solution
set of the equation 𝑥 squared is equal to three 𝑥 plus 10 is negative two,
five.
In our final example, we will
consider what happens when we want to solve 𝑓 of 𝑥 equal to a constant other than
zero.
The graph shows the function 𝑓
of 𝑥 is equal to two 𝑥 squared minus four 𝑥 minus six. What is the solution set of 𝑓
of 𝑥 equals zero? What is the solution set of 𝑓
of 𝑥 equals negative six?
The solution set of 𝑓 of 𝑥
equals zero is the set of values of 𝑥 for which the curve crosses the 𝑥-axis,
since 𝑦 equals zero at these points. Inspecting the graph, we can
see that the curve crosses the 𝑥-axis at two points: 𝑥 equals negative one and
𝑥 equals three. Therefore, the solution set of
𝑓 of 𝑥 equals zero is negative one, three.
Just as the solution of 𝑓 of
𝑥 equals zero is the set of values for which 𝑦 equals zero, the solution set
of 𝑓 of 𝑥 equals negative six is the set of values where 𝑦 equals negative
six. We can find this by drawing a
horizontal line at 𝑦 equals negative six and determining the coordinates of the
points where the line and curve intersect. We can identify from the graph
that the points of intersection are at 𝑥 equals zero and 𝑥 equals two. Therefore, the solution set of
𝑓 of 𝑥 equals negative six is zero, two.
We will now finish this video by
recapping the key points.
Solving quadratic equations
graphically is often an easier and faster method than doing so algebraically. To solve a quadratic equation
graphically, we inspect the graph to find the points where the graph crosses the
𝑥-axis. We may have two, one, or no
solutions depending on whether the function crosses the 𝑥-axis twice, once, or
never. We can solve quadratic equations
formed by setting 𝑦 equals 𝑑 by drawing a horizontal line at the value 𝑦 equals
𝑑 and by finding where it intersects the function.
