Completing the Square

by | Jan 12, 2021 | Math Learning


A brief discussion about what completing the square is and what we use it for. Focuses on using the technique for other reasons than solving equations.

Background Math

Development of the patterns we use for completing the square. This part is important since it introduces the notation we will be using!

Introductory Examples

The first three examples step through the process very slowly with a great deal of explanation for each step. These are intended for people who are unfamiliar with the process and (perhaps) need to take the algebra slowly.

Note that the three examples increase in difficulty.

Example 1: How to Complete the Square

Example 2: A Little More Complicated Example

Example 3: Dealing With Leading Coefficients

Fast Example

Example 4 is similar in difficulty to Example 3, but worked through quickly and with less explanation.

The Quick Method

With a solid grounding in the ideas and methods of completing the square, this discussion shows how we can cut out most of the background steps and basically just do things in one step (at least for the easy ones).

Quick Method Examples

Demonstrating the Quick Method.

Example 5: Simple example with a leading coefficient of 1.

Example 6: Quick method when the leading coefficient is not 1.

Solving Equations by Completing the Square

An example and discussion of solving a quadratic equation by Completing the Square



What Is Completing the Square?

Completing the Square is a technique in algebra that allows us to rewrite a quadratic expression that is in standard form in vertex form. That is

so that ax2+bx+c=a(x−h)2+k.

Why Do We Complete the Square?

It should be noted that many times, students think (are taught?) that completing the square is just a way to solve a quadratic equation that has the form:

However, in other areas of mathematics, we sometimes need to express a quadratic in the vertex form. The following are two examples of this.

Please note, you don’t have to understand these examples! That’s not the point of them!


\item \textit{Integral Calculus:}

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\item \textit{Differential Equations (Laplace Transforms):}

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The two examples above are \textit{only} to show you that completing the square is used for \textit{other reasons} than to solve equations.\vskip 1.5cm

\textbf{\large Background (Leading up to \textit{How To\ldots})}

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Suppose we have a perfect square binomial that we expand using FOIL:


If we reverse this string of equations, we see that any quadratic that has this pattern: x2+2Bx+B2 is a perfect square and we can factor it as (x+B)2.

\vskip 2mm

To Complete the Square, we adjust a quadratic expression so that it exhibits the perfect-square pattern. Here is an overview of the rest of the article so you can focus on what you need most.\vskip 1.5cm

\textbf{\large Introductory Examples}\vskip 2mm

\textbf{Example 1 How To Complete the Square}\vskip 2mm

Complete the square on x2+14x+40.

\vskip 5mm

\textbf{Solution}\vskip 2mm

Match up the quadratic expression with the perfect square pattern, starting with the x2 term.

Perfect-Square Pattern:Our Quadratic:x2+2Bx+B2x2+14x+40

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Step 1: Matching The Quadratic Terms\vskip 2mm

The quadratic terms match since they have the same coefficient.

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Step 2: Matching the Linear Terms\vskip 2mm

The \textit{value} of 2B is assigned be us and is whatever we need it to be so we can match the perfect-square pattern. In this example, they match, if we assign 2B=14.

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\vskip 5mm

Step 3: Matching the Constant Term\vskip 2mm

In order to match the perfect square pattern, the constant term has to be equal to B2. Since we defined 2B=14 in Step 2, we know that B=7 which implies B2=72=49.

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To adjust our constant term, we add zero (which won’t change the value of anything) and choose to rewrite the zero as B2–B2.

x2+14x+40=x2+14x+0+40=x2+14x+(49–49)+40=x2+14x+49–49+40=(x2+14x+49)–49+40Adding 0.Since B2–B2=49−49=0Regrouping

Matching up to the pattern:

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Since the first three terms (the ones in the parentheses) match the perfect-square pattern, we know those three terms will factor into (x+B)2.

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\textbf{Answer:} x2+14x+40=(x+7)2–9

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\textbf{Example 2 (A \textit{Little} More Complicated)}\vskip 2mm

Complete the square on x2–9x+3.

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\textbf{Solution} As before, we match up our quadratic with the perfect-square pattern. We’ll do this one a bit more quickly.\vskip 2mm

Step 1: Quadratic and Linear Terms

Both quadratic terms have a coefficient of 1 so they already match. The linear terms will match if we assign 2B=−9. So without any real work or effort we match the first two terms already.

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Step 2: Constant Term

Since we assigned 2B=−9, we divide by 2 to get B=−92. And we know our constant term needs to be

and we introduce it the same way we did before, by adding zero.

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Making sure we match the pattern, we have

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Since the three terms in the parentheses match the perfect-square pattern, we know those three terms factor as a perfect square.

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\textbf{Answer:} x2–9x+3=(x–92)2–694

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\textbf{Example 3: Dealing With Leading Coefficients}\vskip 2mm

Complete the square on 2×2+12x–5.

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\textbf{Solution} In order to complete the square, we first need to reduce the leading coefficient to 1.

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Step 1: Factor out the leading coefficient out of the first two terms.

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Step 2: Complete the Square \textit{Inside the Braces}\vskip 2mm

\qquad Quadratic Term: \textit{Inside} the braces, our quadratic term has a coefficient of 1.\vskip 2mm

\qquad Linear Term: \textit{Inside} the braces, we set 2B=6.\vskip 2mm

A quick check and we see that we are matching the perfect-square pattern \textit{inside the braces}.

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Let’s move on to the constant term.\vskip 2mm

\qquad Constant Term: \textit{Inside} the braces we have 2B=6 which means B=3 and so B2=32=9. As before, adjust our constant term by adding {\color{red} 0}, but now it is \textit{inside the braces}.

2[x2+6x]–5=2[x2+6x+0]–5=2[x2+6x+(9–9)]–5=2[x2+6x+9–9]–5=2[(x2+6x+9)–9]–5Adding 0 \textit{inside} the bracesSince B2–B2=9–9=0Regrouping \textit{inside} the braces

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Still keeping our attention focused \textit{inside} the braces, we check to see if we match the perfect-square pattern.

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Since the three terms inside the parentheses match the pattern, we know those three terms form a perfect square, so they will easily factor into (x+B)2.

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Step 3: Adjusting the Final Form\vskip 2mm

Start by multiplying the leading coefficient \textit{through the square braces} (NOT the parentheses!), and simplify.

Missing \end{align*}

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\textbf{Answer:} 2×2+12x–5=2(x+3)2–23

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\textbf{\large Quick Example}\vskip 2mm

\textbf{Example 4: A Quick Example}\vskip 2mm

Complete the Square on 3×2–4x+8.\vskip 5mm

\textbf{Solution:}\vskip 2mm

Missing \end{align*}

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\textbf{Answer:} 3×2–4x+8=3(x–23)2+203

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\textbf{\large The Quick Method}\vskip 2mm

The method shown in the examples can be made more efficient if we recognize that the pattern is always the same. For a simple quadratic with a leading coefficient of 1, the completed square form looks like this:

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Inside the final parentheses we always end up with x+B, where B is half of the coefficient of the original x term.\vskip 2mm

Next, we subtract B2 \textit{outside} the parentheses. Let’s try it with one of our previous examples to see it in action.\vskip 5mm

\textbf{Example 5: Simple Quick Method}\vskip 2mm

Use the quick version of Completing the Square on x2+14x+40.

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\vskip 5mm

\textbf{Answer:} x2+14x+40=(x+7)2–9

Note: This is the same answer we got in Example 1.\vskip 1.5cm

\textbf{Example 6: Quick Method with Leading Coefficient}\vskip 2mm

The quick version can also be used when the leading coefficient isn’t 1. Like before, we just factor out the leading coefficient first, and then work inside the braces.\vskip 2mm

Let’s complete the square on 5×2+80x+13.

\noindent\textbf{Solution}\vskip 2mm

Step 1) Factor out the leading coefficient.

Step 2) Now use the quick method to complete the square on the \textit{inside} of the braces.

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\textbf{Answer:} 5×2+80x+13=5(x+8)2–307

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\textbf{\large Solving Equations}\vskip 2mm

When solving a quadratic equation, I find the technique of completing the square is not very efficient, except in the simple cases where the leading coefficient is 1 and the linear coefficient is even.\vskip 2mm

But that doesn’t mean we shouldn’t know \textit{how} to solve more complicated quadratic equations with this technique.\vskip 5mm

\textbf{Two Approaches}\vskip 2mm

When solving an equation through completing the square, there are two basic approaches: (1) Complete the square first, \textit{then} solve the equation, or (2) Solve the equation \textit{as} you complete the square.\vskip 5mm

\textbf{Example 7: Solving a Quadratic Equation by Completing the Square}\vskip 2mm

Use Completing the Square to solve 4×2+3x–10=0.\vskip 5mm

\textbf{Solution (Method 1)}\vskip 2mm

For this approach, we’ll first complete the square, then solve the equation.\vskip 2mm

Step 1) Complete the Square on 4×2+3x–10.\vskip 2mm

Using the techniques we’ve discussed above, we get the following.

\vskip 5mm

Step 2) Solve the equation.

4×2+3x–104(x+38)2–169164(x+38)2(x+38)2x+38x+38x=0=0=16916=16964=±16964−−−−√=±138=–38±138Complete the SquareAdd the constantDivide by 4Take the square rootSimplifySubtract

The two solutions are

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\textbf{Solution (Method 2)}\vskip 2mm

This time, we’ll complete the square as we solve the equation. It should be noted that this approach is what many students think \textit{is} Completing the Square.\vskip 2mm

4×2+3x–104×2+3xx2+34xx2+34x+964(x+38)2x+38x=0=10=104=104+964=16064+964=±16964−−−−√=−38±138Add the constantDivide by 4Add B2Factor the left sideTake square rootSubtract

Again, we see the two answers are x=−38+138=54 and x=−38–138=−2.

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\textbf{Answer:} x=54 or x=−2

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\textbf{Conclusion}\vskip 2mm

Admittedly, most high school student will Complete the Square in the context of solving quadratic equations. However, it is important that they understand that the technique is more than just an equation-solving technique. Outside of high school math, much of its use is in changing the form of a quadratic function so that it can be used in higher mathematical techniques.

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