Math Lesson 15.2.2 - Using the Completing-the-Square-Method to Plot Quadratic Graphs

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Welcome to our Math lesson on Using the Completing-the-Square-Method to Plot Quadratic Graphs, this is the second lesson of our suite of math lessons covering the topic of Quadratic Graphs Part Two, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Using the Completing-the-Square-Method to Plot Quadratic Graphs

In tutorial 9.5, we explained that some quadratic equations with one variable can be solved by completing the square according to the formula

where a = 1, p = b/2 and q is the remaining part of the original constant c.

The next step involves expressing the new constant as the square of a number, i.e. the constant q in the above formula is the square of a number preceded by a negative sign. In this way, we complete the square. Then, we solve the equation by sending the new constant on the other side. Since it is preceded by a negative sign, it becomes positive when

For example, in the quadratic equation

we first identify the original coefficients and constant (a = 1, b = -6 and c = 8), then we identify the (x + b/2)² part, i.e.

then, we identify the rest of the constant by comparing the new expression with the left part of the original equation, i.e.

Therefore, we can express the original equation as

This equation is true for

and

The first equation gives

Thus, the first root is

and the second equation gives

Thus, the second root is

In this way, using the known form of factorisation for quadratic equations with a = 1

we obtain after substituting the values found above:

Proof: Expanding the last equation yields

which is identical to the original equation.

What if the coefficient a is different from 1? Well, in this case, we must try to write the simplified version of the quadratic equation (without the variable y)

in the form

where

In this way, we obtain

Why did we stop here with this? Well, the objective of this tutorial is to know how to plot the graph of a quadratic equation with two variables having the general form

Thus, after expressing this equation in the form

to complete the square, we find the roots (if any) that indicate the x-intercepts and then, we find the rest of the points as in the other cases discussed so far.

Example 4

Plot the graph of the quadratic equation

Solution 4

First, let's identify the original coefficients and constant. We have a = 3, b = 5 and c = 2. Thus, since the coefficient a is different from 1, we must write the equation in the form

to complete the square. Given that

we obtain after substituting the known values

and

Therefore, we write the equation in the form

We calculate the roots by reducing this equation to the simplified form that contains a single variable, i.e.

Solving this equation yields

This equation is true for

and

Solving the first equation yields

and solving the second equation yields

Hence, the two x-intercepts of the graph are: A(-1, 0) and B(-2/3, 0).

As we said before, the rest of the solution is the same as in the other methods. Thus, first, we find the vertex V. The x-coordinate of this point is (as usual)

The corresponding y-value of the vertex is

Hence, the vertex of the parabola is at V(-5/6, -1/12).

Another special point you already know how to find is the y-intercept (where x = 0). Thus, substituting x = 0 in the original equation we obtain y = c = 2. Therefore, point C(0, 2) is another point on the graph.

The last point we need to plot the graph is point D, which is symmetrical to point C concerning the vertical line drawn from the vertex. This point will have the same y-coordinate as C as it is symmetrical, so we have y_D.

As for the x-coordinate of point D, we use the half-segment formula for the horizontal direction, where one of the endpoints of the segment is x_C, the midpoint is x_V and the other endpoint (to be found) is x_D. Thus, we have

Hence, point D is at D(-5/3, 2).

Inserting all the above five points in the coordinates system and connecting them smoothly gives the graph of the equation y = 3x² + 5x + 2, as shown in the figure below.

You have reached the end of Math lesson 15.2.2 Using the Completing-the-Square-Method to Plot Quadratic Graphs. There are 2 lessons in this physics tutorial covering Quadratic Graphs Part Two, you can access all the lessons from this tutorial below.

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