Formula, Steps, Solving & Examples (2023)

When dealing with algebraic expressions, it is always helpful to view them in their simplest form. That way, we can solve these expressions easily and determine possible patterns involved. In this case, we want to look at simplifying quadratic equations. So far, we have learned factoring methods such as grouping and identifying the greatest common factor. In this article, we shall be introduced to a new concept called completing the square. We will see the steps for solving quadratic equations by completing the square and examples of its application.

Complete Squares

Expressions of the formare known as a complete square.

Expanding the first expression, we obtain

.

Similarly, the second expression becomes

,

upon expansion.

Completing the square is a method used to simplify a quadratic equation into an algebraic expression that can be easily solved. This technique can also be used to determine the maximum or minimum values of a quadratic equation and help with graphing.

The aim here is to convert the standard form of a quadratic equation so that it looks like the expression above. The general formula for completing the square is as follows.

Geometrical Representation of Completing the Square

So what does it mean to complete the square? Before we get into some examples involving quadratic equations, it may be helpful to understand the geometry behind this method. Let us observe the diagram below.

Completing the square, Aishah Amri - StudySmarter Originals

In the first image, we have the red square and the green rectangle. Adding these two shapes together, we obtain the expression

.

We want to rearrange this so that it looks like a square. Halving the width of the green rectangle, we obtain.

Now rearranging these two new smaller green rectangles, we have the second image. Notice that we have a missing segment at the corner of the second image. Thus, to complete this square, we need to add the area of the blue square,. The complete square is shown in the third image. We can represent this algebraically as follows.

where the termcompletes the square.

Completing the Square for Quadratic Equations of the Form x²+ bx+ c

Here, we will deal with quadratic equations of the form x² + bx + c, where the coefficient of the term x² is 1.

Balancing the Equation

From the previous section, it is important to note that we cannot add the term without subtracting it from our initial expression. If we do not do this, our equation changes entirely. Let us show this with the following example.

Complete the square of the quadratic equation .

Solution

From the above equation, we have b = 4 and c = 3.

To complete the square, we have

Notice that we have to add and subtract to balance the equation. Now, observe that the first three terms above can be simplified into the form of a completed square as

Thus, the equation becomes

As you can see, we've changed the equation from the standard form to vertex form with h = –2 and k = –1.

Solving Completed Square Expressions

From the example above, observe that the variable x only appears once in the completed square expression. This allows us to solve the equation for x using basic algebra. Additionally, this means that we do not have to go through the hassle of performing the Quadratic Formula to solve a given quadratic equation. Let us return to the previous example.

A Quicker Way to Complete the Square

In some cases, the method above can be difficult to solve, especially when we are given quadratic equations with larger coefficients. It is always helpful to know a shortcut to make calculations faster. By completing the square, we want our expression to take the vertex form:

.

Expanding the first term of this expression yields the definition of a complete square. The equation now becomes,

.

Complete the square of the following quadratic equation:

.

Solution

Given the quadratic equation above, use the expression to express the left-hand side as .

Aligning these two expressions, we have

From here we can see that 4x must take the form of 2hx, so d = 2 since

The constant 3 must take the form h² + k and since we know that d = 2,

Thus, we obtain

as we have solved before.

Use the complete the square method to solve the quadratic equation .

Solution

From the above equation, we have b = –6 and c = –12.

We then note that

Simplifying the first three terms as a complete square and solving, we obtain

Now solving for x

Thus we have two solutions, correct to two decimals

Completing the Square for Quadratic Equations of the Form ax²+ bx+ c

In this section, we will deal with quadratic equations of the form ax² + bx + c, where the coefficient of the term x² is not equal to 1. For this form of quadratic equation, we can follow the steps below to complete the square.

Completing the Square

Step 1: Given the standard form of the quadratic equation divide all terms by a, that is the coefficient of x²

.

Step 2: Move the term to the right-hand side of the equation.

.

Step 3: Complete the square on the left-hand side of the equation. Balance the equation by adding the same value to the right-hand side

Step 4: Take the square root of both sides

Step 5: Solve to find x

Notice that we have derived the Quadratic Formula using the method of completing the square!

Now going back to Step 3, we have deduced that

Bringing the terms from the right-hand side back to the left-hand side

Multiplying the entire equation by a and simplifying, we obtain

Observe that the equation now takes the form where

which is exactly the general form of completing the square for a quadratic equation as we have initially mentioned in the beginning. Below are some worked examples that demonstrate this.

Complete the square of and solve for x.

Solution

Step 1: Divide the expression by a = 10

Step 2: Move the term to the other side

Step 3: Complete the square and balance the equation

Now multiplying the entire equation by a = 10, we obtain the vertex form

Step 4: Taking the square root on both sides

Note: Remember that

Step 5: Solving for x,

Thus, we have two solutions

Complete the square of and solve for x.

Solution

Step 1: Divide the expression by a = –3

Step 2: Move the term to the other side

Step 3: Complete the square and balance the equation

Now multiplying the entire equation by a = –3, we obtain the vertex form

Step 4: Taking the square root on both sides

Step 5: Solving for x,

Therefore, we have two solutions

Identifying the Maximum and Minimum Values of a Quadratic Equation

Completing the square also helps us determine the maximum and minimum values of a given quadratic equation. By doing so, we can locate this value and plot the graph of a quadratic equation more accurately.

For the general form of a quadratic equation, the maximum and minimum values on a graph take on the following two conditions.

A general plot of the maximum and minimum values of a quadratic equation, Aishah Amri - StudySmarter Originals

Essentially, if the coefficient of x² is positive, then the graph curves downwards and if the coefficient of x² is negative, then the graph curves upwards. From the general formula of completing the square, when the coefficient of x² is 1,

,

the x and y coordinates of the turning point, or the vertex, can be found by the point (h, k). Similarly, when the coefficient of x² is not 1,

,

the x and y coordinates of the turning point, or the vertex, can be found by the same point, (h, k). Note that the value of a does not affect the position of the vertex!

Let us look for the maximum and minimum values for the last two examples from the previous section.

Determine whether the quadratic equation has a maximum or minimum value. Hence, find the coordinates of its turning point.

Solution

The coefficient of the term x² is positive, as a = 10. Thus, we have a minimum value. In this case, the curve opens up. From the derivation of the completed square form of this expression, we obtain

.

Here, .

Remember that the value of a does not vary the x-value of the vertex!

Thus, the minimum value is when .

The coordinates of the minimum turning point is The graph is shown below.

Determine whether the quadratic equation has a maximum or minimum value. Hence, find the coordinates of its turning point.

Solution

The coefficient of the term x² is negative, as a = –3. Thus, we have a maximum value. In this case, the curve opens down. From the derivation of the completed square form of this expression, we obtain

.

Here, .

Thus, the maximum value is when .

The coordinates of the maximum turning point is The graph is shown below.