## Navier-Stokes is hard?

Navier-Stokes is hard?
Author: James Lowman}

I was recently asked the question; “\textbf{How can the Navier-Stokes equations both describe our observable world and not be known to always have solutions in 3D?}”

For those that don’t know, the Navier-Stokes (NS) equations represent a mathematical model that describes fluid.
It is derived from quantities in physics that are conserved.
Those are:
\begin{itemize}
\item Conservation of mass
\item Conservation of momentum
\item Conservation of energy
\end{itemize}

The Navier-Stokes equations are yet to be considered solved.
There is a prize association called “The Millennium Prize Problems” which offers one million dollars to anyone that can solve one of the seven problems they have listed, and the existence and smoothness of a solution to the Navier-Stokes equations is on that list.

So the question I was asked seems completely reasonable.
How can these equations describe physical phenomena with knowledge of a solution?
The answer touches on an interesting, and modern, intersection in science.

We use a number of assumption to simplify a NS system in order to approximate a solution.
These approximations are usually handled by a computational fluid dynamics software package.
They are high powered suites that allow for customized physics simulations.
The assumptions most often used to simplify the NS equations are things like the fluids being in-compressible, the turbulence affects are averaged, or that the fluid is Newtonian.
Suffice to say, even with a high powered computational engine, more often then not a scaled down and simplified version of the NS equations are all that we can easily approximate.

So if the solutions to the fluid problems are all approximate, how can we know if they are describing real physics?
To compare a NS system solution to an actual physical fluid flow, we would need to perform an experiment and collect data on such a flow.
This results in flawed data, because the act of introducing measurement tools into fluid domains involves disrupting the fluid.
The best measurements of fluid dynamics themselves are approximations.

So what we have ended up with is approximate mathematical approximations of fluids approximating approximate experiments.
Its not hard to see why the student’s question needed clarification.

The answer is: \textbf{Because it works}.

That’s a pretty hand-waved explanation, that no student should accept.
But for anyone interested, the derivation of Navier-Stokes is a fun derivation that follows from those conservation laws that were listed, and experiments can be found in open source publications that show how much effort was placed on collecting data with minimal interference.
The interested party might be able discern that experimental data matches model solutions (reasonably well), and we can start to believe that Navier-Stokes delivers on the promise of predicting real world physics.

## The Quadratic Equation

James Lowman
July 2019
The quadratic equation is always the answer.
Time and time again, while tutoring students, I encounter a resistance to
using an unfailing tool called the quadratic equation. This simple algebraic
mathematical statement allows a student to find the roots of any clumsy second
order polynomial with ease. I can only assume that, while the quadratic equation
is drilled into memory, its useful nature is under-reported in high school mathematics.
For those who aren’t familiar:
$ax2+bx+c= 0$                                     (1)
x
=
b
±
b
2
4
ac
2
a
[/Latex]
(2)
If you can get an equation into the form of equation
??
, then you can use the
coefficients
a
,
b
, and
c
in order to solve for
x
in equation
??
. The
±
in equation
??
implies that there are two solutions for
x
(this being the result of solving a
second order polynomial equation). These two solutions represent the roots of
equation
??
. The roots are the values of
x
that allow equation
??
to equal 0 on
the left hand side.
Roots are of interest to us, because they are a quick and easy way to
determine factors of a quadratic (not to mention all the mathematical reasons
to identify when a function is equal to zero). Students are drilled on factoring.
I rarely see a student that is unable to factor a simple quadratic, but I often
encounter those that have problems when the factors fail to be immediately
obvious. The quadratic equation is always the answer.
I often despair that education relies entirely on problem singularity, a belief
that there is only one correct way to approach a question. When I see a student
struggle for minutes to try and factor a difficult quadratic, I can’t help but
wonder why they shy away from the quadratic equation. Sometimes it is hyper-
focus that keeps them from finding an alternate path forward, but more and
more I encounter fear. Factoring is supposed to be an easy shortcut, while the
quadratic equation has the confusing
±
and the scary
. But a shortcut fails
to be short when it takes time to muddle through possible variations. Take, for
example, the following quadratic:

## Cardinality and Countably Infinite Sets

Cardinality is a term used to describe the size of sets. Set A has the same cardinality as set B if a bijection exists between the two sets. We write this as |A| = |B|. One important type of cardinality is called “countably infinite.” A set A is considered to be countably infinite if a bijection exists between A and the natural numbers ℕ. Countably infinite sets are said to have a cardinality of אo (pronounced “aleph naught”).

Remember that a function f is a bijection if the following condition are met:

1. It is injective (“1 to 1”): f(x)=f(y)⟹x=y

2. It is surjective (“onto”): for all b in B there is some a in A such that f(a)=b.

A set is a bijection if it is both a surjection and an injection.

Example 1. Show that the set of integers ℤ is countably infinite.

To show that ℤ is countably infinite, we must find a bijection between ℕ and ℤ, i.e. we need to find a way to match up each element of ℕ to a unique element of ℤ, and this function must cover each element in ℤ.

We can start by writing out a pattern. One pattern we can use is to count down starting at 0, then going back and “picking up” each positive integer. This follows the pattern {0, -1, 1, -2, 2, -3, 3…}. We match to ℕ to ℤ as follows:

 ℕ 0 1 2 3 4 … ℤ 0 -1 1 -2 2 …

Notice that each even natural number is matched up to it’s half. It follows the function f(n) = n/2.

The odd numbers follow the function f(n) = -(n+1)/2

We can write this as a piecewise function as:

$f(n) = \begin{cases} n/2, & \text{if n is even} \\ -(n+1)/2, & \text{if n is odd} \end{cases}$

Now we need to check if our function is a bijection.

Injectivity: Suppose the function is not injective. Then there exists some natural numbers x and y such that f(x)=f(y) but x≠y.

For even integers, x/2 = y/2 ⟹ x=y

For odd integers, Then (-x+1)/2 = -(y+1)/2 ⟹ x=y

These are contradictions, so the function is injective.

Surjectivity: Suppose the function is not surjective. Then there is some integer k such that there is no n in ℕ for which f(n) = k.

For k≥0, k=n/2 ⟹2k=n ⟹ n is an even number in ℕ

For k<0, k=-(n+1)/2 ⇒ -2k-1 = n ⇒ n is an odd number in ℕ

These are contradictions, so the function is surjective.

Since f is both injective and surjective, it is a bijection. Therefore, |ℤ| = |N| =אo. ∎

This result is often surprising to students because the set ℕ is contained in the set ℤ.

Example 2. For A = {2n | n is a number in ℕ}, show that |A| = |ℤ| (the set of integers has the same cardinality as the set of even natural numbers).

We can either find a bijection between the two sets or find a bijection from each set to the natural numbers. Since we already found a bijection from ℤ to ℕ in the previous example, we will now find a bijection from A to ℕ.

One function that will work is f(n) = n/2. Checking that it is a bijection is very similar to Example 1.

Since |A| = |ℕ| and |ℤ| = |ℕ|, then |A| = |ℤ| = אo.∎

There are many sets that are countably infinite, ℕ, ℤ, 2ℤ, 3ℤ, nℤ, and ℚ. All of the sets have the same cardinality as the natural numbers ℕ. Some sets that are not countable include ℝ, the set of real numbers between 0 and 1, and ℂ.

Georg Cantor was a pioneer in the field of set theory and was the first to explore countably infinite sets

# Completing the Square

### HT Goodwill

• Introduction

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

• Conclusion

### Introduction

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

$% ax^2 + bx + c\quad \underset{\mbox{\scriptsize the Square}}{\xrightarrow{\mbox{\scriptsize Complete}}}\quad a(x – h)^2 + k$

so that $$ax^2 + bx + c = a(x-h)^2 + k$$.

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:

$% ax^2 + bx + c = 0.$

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!
\begin{itemize}
\item \textit{Integral Calculus:}
\begin{center}
\begin{tikzpicture}
\node [align=center] at (-1,0.75) {\scriptsize Cannot\-2mm]\scriptsize Integrate}; \node [align=center] at (1,0.75) {\scriptsize Can Now\\[-2mm] \scriptsize Integrate}; \node at (0,0) {$$\int \frac{dx}{x^2 + 4x + 5} = \int \frac{dx}{(x+2)^2 + 1}$$}; \draw [-latex] (-1,-0.3) arc [start angle=-180, end angle=0, x radius=1cm, y radius=0.5cm]; \node at (0,-1) [anchor=north,inner sep=0pt,align=center] {\scriptsize Complete the Square\\[-1mm] \scriptsize in Denominator}; \node at (3,-2) [anchor=east] {\tiny www.mathtutoringacademy.com}; \end{tikzpicture} \end{center} \item \textit{Differential Equations (Laplace Transforms):} \begin{center} \begin{tikzpicture} % \draw[help lines,step=5mm] (-2,-1) grid (2.5,1); \node [align = center] at (-1.25,1) {\scriptsize Cannot Find\\[-2mm] \scriptsize Transform}; \node [align=center] at (1.25,1) {\scriptsize Can Now Find\\[-2mm] \scriptsize Transform}; \node at (0,0) {$$\mathcal{L}^{-1}\left\{\frac{5}{s^2 + 4s + 5}\right\} = \mathcal{L}^{-1}\left\{\frac{5}{(s+2)^2 + 1}\right\}$$}; \draw [-latex] (-1.25,-0.3) arc [start angle=-180, end angle=0, x radius=1.5cm, y radius=0.5cm]; \node at (0,-1) [anchor=north,inner sep=0pt,align=center] {\scriptsize Complete the Square\\[-1mm] \scriptsize in Denominator}; \node at (3,-2) [anchor=east] {\tiny www.mathtutoringacademy.com}; \end{tikzpicture} \end{center} \end{itemize} 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})} \vskip 2mm Suppose we have a perfect square binomial that we expand using FOIL: \begin{align*} (x + B)^2 & = (x + B)(x + B)\\ & = x^2 + Bx + Bx + B^2\\ & = x^2 + 2Bx + B^2 \end{align*} If we reverse this string of equations, we see that any quadratic that has this pattern: $$x^2 + 2Bx + B^2$$ is a perfect square and we can factor it as $$(x + B)^2$$. \[% \underbrace{x^2 + 2Bx + B^2}_{\mbox{\parbox{3cm}{\centering \scriptsize The Perfect-Square\\Pattern\ldots}}} = \underbrace{(x+B)^2}_{\mbox{\parbox{3cm}{\centering\scriptsize \ldots always factors\\this way.}}}\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 $$x^2 + 14x + 40$$.
\vskip 5mm

\textbf{Solution}\vskip 2mm

Match up the quadratic expression with the perfect square pattern, starting with the $$x^2$$ term.

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + 2Bx + B^2\\
\mbox{Our Quadratic:} & \quad x^2 + 14x + 40
\end{align*}
\vskip 5mm

Step 1: Matching The Quadratic Terms\vskip 2mm

The quadratic terms match since they have the same coefficient.

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + 2Bx + B^2\\
\mbox{Our Quadratic:} & \quad x^2 + 14x + 40\\
\mbox{Match?} & \quad \checkmark
\end{align*}
\vskip 5mm

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 $${\color{blue} 2B = 14}$$.

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + {\color{blue} 2B}x + B^2\\
\mbox{Our Quadratic:} & \quad x^2 + {\color{blue}14}x + 40\\
\end{align*}
\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 $$B^2$$. Since we defined $$2B = 14$$ in Step 2, we know that $$B = 7$$ which implies $${\color{red} B^2 = 7^2 = 49}$$.
\vskip 2mm

To adjust our constant term, we add zero (which won’t change the value of anything) and choose to rewrite the zero as $$B^2 – B^2$$.

\begin{align*}
x^2 + 14x + 40 & = x^2 + 14x + {\color{red} 0} + 40 &&\mbox{Adding }{\color{red} 0}.\\
& = x^2 + 14x + {\color{red} (49 – 49)} + 40 &&\mbox{Since } B^2 – B^2 = {\color{red} 49-49 = 0}\\
& = x^2 + 14x + 49 – 49 + 40\\
& = (x^2 + 14x + 49) – 49 + 40&&\mbox{Regrouping}
\end{align*}

Matching up to the pattern:

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + 2Bx + B^2\\
\mbox{Our Quadratic:} & \quad (x^2 + 14x + 49) – 49 + 40\\
\end{align*}

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

\begin{align*}
{\color{blue}\overbrace{(x^2 + 14x + 49)}^{\mbox{Perfect Square}}} – 49 + 40
& = {\color{blue}\overbrace{(x+7)^2}^{\mbox{Factored}}} {\color{red}- 49 + 40}\\
& = (x+7)^2 {\color{red}- 9}
\end{align*}

\textbf{Answer:} $$x^2 + 14x + 40 = (x+7)^2 – 9$$
\vskip 1.5cm

\textbf{Example 2 (A \textit{Little} More Complicated)}\vskip 2mm
Complete the square on $$x^2 – 9x + 3$$.
\vskip 5mm

\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 $${\color{blue}2B = -9}$$. So without any real work or effort we match the first two terms already.

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + {\color{blue} 2B}x + B^2\\
\mbox{Our Quadratic:} & \quad x^2\,{\color{blue}-\,9}x + 3\\
\end{align*}

Step 2: Constant Term

Since we assigned $${\color{blue} 2B = -9}$$, we divide by 2 to get $$B = -\frac 9 2$$. And we know our constant term needs to be
$% {\color{red}B^2 = \left(-\frac 9 2\right)^2 = \frac{81} 4},$

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

\begin{align*}
x^2 – 9x + 3
& = x^2 – 9x + {\color{red} 0} + 3 && \mbox{Adding }{\color{red} 0}\\
& = x^2 – 9x + {\color{red}\left(\frac{81} 4 – \frac{81} 4\right)} + 3 && \mbox{Since } B^2 – B^2 = {\color{red} \frac{81} 4-\frac{81} 4 = 0}\\
& = x^2 – 9x + \frac{81} 4 – \frac{81} 4 + 3\\
& = \left(x^2 – 9x + \frac{81} 4\right) – \frac{81} 4 + 3 && \mbox{Regrouping}
\end{align*}

Making sure we match the pattern, we have

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad\quad x^2 + 2Bx + B^2\\
\mbox{Our Quadratic:} & \quad \left(x^2 + 14x + \frac{81} 4\right) – \frac{81} 4 + 40\\
\end{align*}

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

\begin{align*}
{\color{blue}\overbrace{\left(x^2 – 9x + \frac{81} 4\right)}^{\mbox{Perfect Square}}} {\color{red}- \frac{81} 4 + 3}
& = {\color{blue} \overbrace{\left(x – \frac 9 2\right)^2}^{\mbox{Factored}}} {\color{red} – \frac{81} 4 + \frac{12} 4}\\
& = \left(x – \frac 9 2\right)^2 {\color{red} – \frac{69} 4}
\end{align*}
\vskip 5mm

\textbf{Answer:} $$x^2 – 9x + 3 = \left(x – \frac 9 2\right)^2 – \frac{69} 4$$
\vskip 1.5cm

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

Complete the square on $$2x^2 + 12x – 5$$.
\vskip 5mm

\textbf{Solution} In order to complete the square, we first need to reduce the leading coefficient to 1.
\vskip 2mm

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

$% {\color{blue} 2}x^2 + 12x – 5 = {\color{blue} 2}[x^2 + 6x] – 5$
\vskip 5mm

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 $${\color{blue} 2B = 6}$$.\vskip 2mm

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

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad\,\,\,\, x^2 + {\color{blue}2B}x + B^2\\
\mbox{Our Quadratic (Inside Braces):} & \quad 2[x^2 + {\color{blue}6}x] – 5\\
\end{align*}

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

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

\begin{align*}
2[x^2 + {\color{blue} 6}x] – 5
& = 2[x^2 + {\color{blue} 6}x + {\color{red} 0}] – 5 && \mbox{Adding } {\color{red} 0} \mbox{ \textit{inside} the braces}\\
& = 2[x^2 + {\color{blue} 6}x + {\color{red} (9 – 9)}] – 5 && \mbox{Since } B^2 – B^2 = {\color{red} 9 – 9 = 0}\\
& = 2[x^2 + 6x + 9 – 9] – 5\\
& = 2[(x^2 + 6x + 9) – 9] – 5 && \mbox{Regrouping \textit{inside} the braces}
\end{align*}
\vskip 2mm

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

\begin{align*}
\mbox{Perfect-Square Pattern:} & \quad x^2 + 2Bx + B^2\\
\mbox{Our Quadratic (Inside Braces):} & \quad 2[(x^2 + 6x + 9) – 9] – 5\\
\end{align*}

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 $$(x + B)^2$$.

\begin{align*}
2\big[{\color{blue}\overbrace{\left(x^2 + 6x + 9\right)}^{\mbox{Perfect Square}}} – 9\big] – 5
& = 2\big[{\color{blue}\overbrace{\left(x + 3\right)^2}^{\mbox{Factored}}} – 9\big] – 5
\end{align*}
\vskip 5mm

Step 3: Adjusting the Final Form\vskip 2mm

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

\begin{align*}%
{\color{blue} 2}\big[\left(x + 3\right)^2 – 9\big] – 5
& = \big[{\color{blue} 2}(x+3)^2 – {\color{blue} (2)} 9\big] – 5 && \mbox{Multiply through braces}\\
& = \big[2(x + 3)^2 – 18\big] – 5\\
& = 2(x + 3)^2 – 18 – 5 && \mbox{Braces no longer needed}\\
& = 2(x+3)^2 – 23 && \mbox{Combine the constants}
\end{align*}
\vskip 5mm

\textbf{Answer:} $$2x^2 + 12x – 5 = 2(x+3)^2 – 23$$
\vskip 1.5cm

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

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

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

\textbf{Solution:}\vskip 2mm

\begin{align*}%
3x^2 – 4x + 8 & = 3\left[x^2 – \frac 4 3 x\right] + 8 && \mbox{Factor out leading coefficient}\6pt] & = 3\left[x^2 {\color{blue}- \frac 4 3} x\right] + 8 && \mbox{Identify } {\color{blue} 2B = -\frac 4 3} \longrightarrow {\color{red} B^2 = \left(-\frac 2 3\right)^2 = \frac 4 9}\\[6pt] & = 3\left[x^2 {\color{blue}- \frac 4 3} x + {\color{red} \left(\frac 4 9 – \frac 4 9\right)}\right] + 8 && \mbox{Add } {\color{red} 0}\\[6pt] & = 3\left[\left(x^2 {\color{blue}- \frac 4 3} x + {\color{red}\frac 4 9}\right)\,\,{\color{red}- \frac 4 9}\right] + 8 && \mbox{Regrouping}\\[6pt] & = 3\left[\left(x – \frac 2 3\right)^2 – \frac 4 9\right] + 8 && \mbox{Factor}\\[6pt] & = 3\left(x – \frac 2 3\right)^2 – \frac 4 3 + 8 && \mbox{Distribute}\\[6pt] & = 3\left(x – \frac 2 3\right)^2 – \frac 4 3 + \frac{24} 3\\[6pt] & = 3\left(x – \frac 2 3\right)^2 + \frac{20} 3 && \mbox{Combine like terms} \end{align*} \vskip 5mm \textbf{Answer:} $$3x^2 – 4x + 8 = 3\left(x – \frac 2 3\right)^2 + \frac{20} 3$$ \vskip 1.5cm \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: \begin{center} \begin{tikzpicture} % Help Lines % \draw [step=0.25cm,help lines] (-2,-1) grid (2,1); % \draw (-2,-1) grid (2,1); % Equation \node {% $$% x^2 + {\color{blue} 2B}x + c = (x + {\color{blue} B})^2 – {\color{blue} B^2} + c$$}; % Arrows \node (linear) [inner sep=0pt] at (-1.625,-0.15){}; \node (BLow) [inner sep=0pt] at (0.75, -0.15){}; \node (BUp) [inner sep=0pt] at (0.75, 0.2){}; \node (B2) [inner sep=0pt] at (1.75,0.2){}; \draw [-latex] (linear.south) to [out=270,in=270] (BLow.south); \draw [-latex] (BUp.north) to [out=90,in=90] (B2.north); % Labels \node at (-0.375, -0.625) {$$\div 2$$}; \node at (1.5, 0.75) {\scriptsize Minus the Square}; \node at (3,-1) [anchor=east] {\tiny www.mathtutoringacademy.com}; \end{tikzpicture} \end{center} 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 $$B^2$$ \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 $$x^2 + 14x + 40$$. \begin{center} \begin{tikzpicture} % Help Lines % \draw [step=0.25cm,help lines] (0,-1) grid (5,1); % \draw (0,-1) grid (5,1); % Equation \node at (1.2,-0.25){% $$% x^2 + {\color{blue} 14}x + 40 = (x + {\color{blue} 7})^2 \underbrace{- {\color{blue} 7^2} + 40}_{\mbox{\scriptsize Add Together}} = (x + 7)^2 – 9$$}; % Arrows \node (linear) [inner sep=0pt] at (-1.625,-0.15){}; \node (BLow) [inner sep=0pt] at (0.75, -0.15){}; \node (BUp) [inner sep=0pt] at (0.75, 0.2){}; \node (B2) [inner sep=0pt] at (1.75,0.2){}; \draw [-latex] (linear.south) to [out=270,in=270] (BLow.south); \draw [-latex] (BUp.north) to [out=90,in=90] (B2.north); \draw [-latex] (2.85,-0.575) .. controls (3,-0.625) and (5,-0.75)..(5,-0.25); % Labels \node at (-0.375, -0.625) {$$\div 2$$}; \node at (1.5, 0.75) {\scriptsize Minus the Square}; \node at (5,-1) [anchor=east] {\tiny www.mathtutoringacademy.com}; \end{tikzpicture} \end{center} \vskip 5mm \textbf{Answer:} $$x^2 + 14x + 40 = (x + 7)^2 – 9$$ 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 $$5x^2 + 80x + 13$$. \noindent\textbf{Solution}\vskip 2mm Step 1) Factor out the leading coefficient. \[% 5x^2 + 80x + 13 = 5[x^2 + 16x] + 13

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

\begin{center}
\begin{tikzpicture}
% Help Lines
% \draw [step=0.25cm,help lines] (-2,-2) grid (4,2);
% \draw (-2,-1) grid (4,1);
% \draw [red, step = 2] (0,-1) grid (3,1);
% Equation
\node at (1.2,0){%
\begin{minipage}{\textwidth}
\begin{align*}%
5[x^2 + {\color{blue} 16}x] + 13
& = 5[x^2 + {\color{blue} 16}x] + 13\5mm] & = 5[(x + {\color{blue} 8})^2 – {\color{blue} 64}] + 13\\[6mm] & = 5(x + {\color{blue} 8})^2 – 320 + 13\\ & = 5(x + {\color{blue} 8})^2 – 307% \end{align*}% \end{minipage} }; % Arrows \node (linear) [inner sep=0pt] at (2.375,0.95){}; \node (BLow) [inner sep=0pt] at (2.23, -0.05){}; \node (BUp) [inner sep=0pt] at (2.23, 0.25){}; \node (B2) [inner sep=0pt] at (3.23,-0.05){}; \draw [-latex] (linear.south) to [out=270,in=90] (BUp.south); \draw [-latex] (BLow.south) to [out=270,in=270] (B2.south); % Labels \node at (2.625, 0.625) {\scriptsize $$\div 2$$}; \node at (3, -0.55) {\scriptsize Minus the Square}; \node at (4.5,-2) [anchor=east] {\tiny www.mathtutoringacademy.com}; \end{tikzpicture} \end{center} \vskip 5mm \textbf{Answer:} $$5x^2 + 80x + 13 = 5(x + 8)^2 – 307$$ \vskip 1.5cm \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 $$4x^2 + 3x – 10 = 0$$.\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 $$4x^2 + 3x – 10$$.\vskip 2mm Using the techniques we’ve discussed above, we get the following. \[% 4x^2 + 3x – 10 = 4\left(x + \frac 3 8\right)^2 – \frac{169}{16}
\vskip 5mm

Step 2) Solve the equation.

\begin{align*}
4x^2 + 3x – 10 & = 0\6pt] 4\left(x + \frac 3 8\right)^2 – \frac{169}{16} & = 0 && \mbox{Complete the Square}\\[6pt] 4\left(x + \frac 3 8\right)^2 & = \frac{169}{16} && \mbox{Add the constant}\\[6pt] \left(x + \frac 3 8\right)^2 & = \frac{169}{64} && \mbox{Divide by $$4$$}\\[6pt] x + \frac 3 8 & = \pm\sqrt{\frac{169}{64}} && \mbox{Take the square root}\\[6pt] x + \frac 3 8 & = \pm\frac{13} 8 && \mbox{Simplify}\\[6pt] x & = – \frac 3 8 \pm\frac{13} 8 && \mbox{Subtract} \end{align*} The two solutions are \[% \begin{array}{rl} x & = \displaystyle – \frac 3 8 + \frac {13} 8 = \frac{10} 8 = \frac 5 4\\[12pt] x & = \displaystyle – \frac 3 8 – \frac{13} 8 = – \frac{16} 8 = – 2 \end{array}
\vskip 5mm

\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

\begin{align*}
4x^2 + 3x – 10 & = 0\\[6pt]
4x^2 + 3x & = 10 && \mbox{Add the constant}\\[6pt]
x^2 + \frac 3 4 x & = \frac{10} 4 && \mbox{Divide by 4}\\[6pt]
x^2 + \frac 3 4 x + \frac 9 {64} & = \frac{10} 4 + \frac 9 {64} && \mbox{Add $$B^2$$}\\[6pt]
\left(x + \frac 3 8\right)^2 & = \frac{160}{64} + \frac 9 {64} && \mbox{Factor the left side}\\[6pt]
x + \frac 3 8 & = \pm \sqrt{\frac{169}{64}} && \mbox{Take square root}\\[6pt]
x & = -\frac 3 8 \pm \frac{13} 8 && \mbox{Subtract}
\end{align*}

Again, we see the two answers are $$x =-\frac 3 8 + \frac{13} 8 = \frac 5 4$$ and $$x = -\frac 3 8 – \frac{13} 8 = -2$$.
\vskip 5mm

\textbf{Answer:} $$x = \frac 5 4$$ or $$x = -2$$
\vskip 1.5cm

\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.