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Completing the Squares

Completing the Squares

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 form(x+a)2 and (x-a)2are known as a complete square.

Expanding the first expression, we obtain

(x+a)2=(x+a)(x+a)=x2+2ax+a2.

Similarly, the second expression becomes

(x-a)2=(x-a)(x-a)=x2-2ax+a2,

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.

For the quadratic equation, ax2+bx+c=0, we can complete the square of this expression by converting this into the form

a(x-h)2+k=0 ,

whereh=b2a and k=c-b24a. This form is known as the vertex form of a quadratic.

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

x2+bx.

We want to rearrange this so that it looks like a square. Halving the width of the green rectangle, we obtainb22.

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,b22. The complete square is shown in the third image. We can represent this algebraically as follows.

x2+bx+b22=x+b22

where the termb22completes 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 b22without 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 x2+4x+3=0.

Solution

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

To complete the square, we have

x2+4x+(42)2+3-(42)2=0

Notice that we have to add 422 and subtract 422 to balance the equation. Now, observe that the first three terms above can be simplified into the form of a completed square as

x2+4x+422=x+422=(x+2)2

Thus, the equation becomes

x2+4x+3=0x2+4x+422+3-422=0(x+2)2-1=0

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.

Solve the equationx2+4x+3=0by completing the square.

Solution

Before, we found that

x2+4x+3=0(x+2)2-1=0

Solving for x, we obtain

(x+2)2=1x+2=±1x=±1-2x=-1-2 and x=1-2

Thus, we obtain two solutions x=-3 and x=-1.

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:

(x-h)2+k.

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

(x-h)2+k=x2-2hx+h2+k.

The trick here is to "force" our given quadratic equation so that it takes the form of the expression above. Let us attempt this method in the previous example.

Complete the square of the following quadratic equation:

x2+4x+3=0.

Solution

Given the quadratic equation above, use the expression x2+2hx+h2+kto express the left-hand side as (x+h)2+k.

Aligning these two expressions, we have

x2+4x+3=0x2+2hx+h2+k=0

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

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

(2)2+e=3e=-1

Thus, we obtain

x2+4x+3=0(x+2)2-1=0

as we have solved before.

Before we move on to the next section, let us show another worked example.

Use the complete the square method to solve the quadratic equation x2-6x-12=0.

Solution

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

We then note that

x2-6x+(62)2-12-(62)2=0

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

x-622-21=0(x-3)2-21=0

Now solving for x

(x-3)2=21x-3=±21x=±21+3x=-21+3 and x=21+3

Thus we have two solutions, correct to two decimals

x=-21+3-1.58 x=21+37.58

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 ax2+bx+c=0, divide all terms by a, that is the coefficient of x2

x2+bax+ca=0.

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

x2+bax=-ca.

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

x2+bax+(b2a)2=-ca+(b2a)2x+b2a2=-ca+b24a2x+b2a2=b2-4ac4a2

Step 4: Take the square root of both sides

x+b2a=±b2-4ac4a2=±b2-4ac4a2x+b2a=±b2-4ac2a

Step 5: Solve to find x

x=±b2-4ac2a-b2ax=-b±b2-4ac2a

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

x+b2a2=-ca+b2a2

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

x+b2a2+ca-b2a2=0

Multiplying the entire equation by a and simplifying, we obtain

ax+b2a2+aca-ab24a2=0ax+b2a2+c-b24a=0

Observe that the equation now takes the form a(x-h)2+k=0 where

h=b2a and k=c-b24a

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 10x2-2x+1=0 and solve for x.

Solution

Step 1: Divide the expression by a = 10

x2-210x+110=0x2-15x+110=0

Step 2: Move the term 110 to the other side

x2-15x=-110

Step 3: Complete the square and balance the equation x2-15x+-12(5)2=-110+-12(5)2x2-15x+1102=-110+1100x-1102=-9100x-1102+9100=0

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

10x-1102+910=0

Step 4: Taking the square root on both sides

x-110=±-9100=±i9100=±3i10

Note: Remember that -1=i, since i2=-1

Step 5: Solving for x,

x=±3i10+110

Thus, we have two solutions

x=-3i+110 and x=3i+110

Complete the square of -3x2-4x+8=0and solve for x.

Solution

Step 1: Divide the expression by a = –3

x2-4(-3)x+8(-3)=0x2+43x-83=0

Step 2: Move the term -83 to the other side

x2+43x=83

Step 3: Complete the square and balance the equation

x2+43x+42(3)2=83+42(3)2x2+43x+232=83+232x+232=289x+232-289=0

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

-3x+232+283=0

Step 4: Taking the square root on both sides

x+23=±289=±273

Step 5: Solving for x,

x=±273-23=±27-23

Therefore, we have two solutions

x=-2(7+1)3 and 2(7-1)3

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.

The vertex is a point at which the curve on a graph turns from decreasing to increasing or from increasing to decreasing. This is also known as a turning point.

The maximum value is the highest point of the curve in a graph. This is also known as the maximum turning point or local maxima.

The minimum value is the lowest point of the curve in a graph. This is also known as the minimum turning point or local minima.

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,

(x-h)2+k=0,

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,

a(x-h)2+k=0,

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 10x2-2x+1=0 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

10x-1102+910=0.

Here, x=110.

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

Thus, the minimum value is 910 when x=110.

The coordinates of the minimum turning point is 110,910. The graph is shown below.

Determine whether the quadratic equation -3x2-4x+8=0 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

-3x+232+283=0.

Here, x=-23.

Thus, the maximum value is 283 when x=-23.

The coordinates of the maximum turning point is -23,283. The graph is shown below.

Completing Squares - Key takeaways

  • For the quadratic equation ax2+bx+c=0, the standard form of the completed square form is a(x-h)2+k=0 where h=-b2a and h=c-b24a
  • Completing the square is a method used to
    1. Simplify a quadratic equation
    2. Determine the maximum or minimum values
  • To complete the square and solve the quadratic equation, we must
    1. Divide the expression by the coefficient of x2
    2. Move the third term to the right-hand side
    3. Complete the square and balance the equation
    4. Take the square roots of both sides
    5. Solve for x
  • If the coefficient of x2 is positive, then we have a minimum value.
  • If the coefficient of x2 is negative, then we have a maximum value.
  • The coordinates for the turning point is (h, k).

Frequently Asked Questions about Completing the Squares

Completing the square is a method used to simplify a quadratic equation and determine the maximum or minimum values

To solve a quadratic equation by completing the square, we must

  1. Divide the expression by the coefficient of x2
  2. Move the third term to the right-hand side
  3. Complete the square and balance the equation 
  4. Take the square roots of both sides
  5. Solve for x

Yes, all quadratic equations can be solved by completing the square 

The formula for completing the square is a(x–d)2+e=0, where d=–b/2a and e=c-b2/4a

To solve a quadratic equation by completing the square, we must

  1. Divide the expression by the coefficient of x2
  2. Move the third term to the right-hand side
  3. Complete the square and balance the equation 
  4. Take the square roots of both sides
  5. Solve for x

Final Completing the Squares Quiz

Question

What is the purpose of completing the square of a quadratic equation?

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Answer

To simplify a quadratic equation and determine the maximum or minimum values

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Question

What are the five steps to solving a quadratic equation by completing the squares?

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Answer

  1. Divide the expression by the coefficient of x2
  2. Move the third term to the right-hand side
  3. Complete the square and balance the equation 
  4. Take the square roots of both sides
  5. Solve for x

Show question

Question

If the coefficient of xis positive in a given quadratic equation, what kind of turning point do we obtain?


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Answer

Minimum value

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Question

If the coefficient of xis negative in a given quadratic equation, what kind of turning point do we obtain?

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Answer

Maximum value

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Question

What is the maximum value? What are other terms that describe this?

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Answer

The maximum value is the highest point of the curve in a graph. This is also known as the maximum turning point or local maxima.


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Question

What is the minimum value? What are other terms that describe this?


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Answer

The minimum value is the lowest point of the curve in a graph. This is also known as the minimum turning point or local minima.

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Question

What is a vertex? Is there another word that can be used to describe this?

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Answer

The vertex is a point at which the curve on a graph turns. This is also known as a turning point.

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Question

For a quadratic equation with a positive term in x2, what do the coordinates (-ad,e) tell us about the graph?

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Answer

The curve has a minimum value of e at x=-ad

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Question

For a quadratic equation with a negative term in x2, what do the coordinates (ad,e) tell us about the graph?

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Answer

The curve has a maximum value of e at x=ad

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Question

Can any quadratic equation be solved by completing the square?

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Answer

Yes, all quadratic equation be solved by completing the square

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