**The 2nd stage of writing a solution : Journeyman**

Review the guidelines for an Apprentice.

Congratulations on writing up solutions! This is an achievement to be celebrated, so give yourself a pat on the back. Over time, it will become easier for you to clearly communicate complicated ideas to others. Remember that practice makes perfect, so keep at it!

As you can see from the solutions of others, there are many different ways to write a solution. Some of them are expressed clearly and quickly get voted up to the top. These are good examples for us to learn from. A clearly presented solution is easier to read and understand. Your audience will appreciate your effort, and will be more likely to finish reading the entire solution and comprehend your wonderful ideas.

## Guidelines

Here are some guidelines for a Journeyman:

**1) Write and explain what you mean.**

When having a monologue with oneself, it is very easy to converse back and forth, because you inherently understand what you want to say. When having a conversation with someone else, this becomes more challenging, as you have to ensure that the other person's doubts are clarified. This becomes difficult when all you can do is write several paragraphs for them to read, and do not have the ability to respond to their concerns through edits or comments thereafter. We are not mind readers, and unless you write down exactly what you mean, your thoughts will remain stuck in your head.

\[ \boxed{ \begin{array} {l }\mbox{If you tell your ideas "I just can't get you out of my head", } \\ \mbox{Everyone else will be "La la la la la la la". } \end{array} } \]

**2) Connect your arguments.**

When we combine various pieces of logic together, we often need to check that they are still valid. If the link between your statements is not strong enough, then your argument is a house of cards. Be aware of the requirements and implications of your arguments.

A common example is that squaring an equation could introduce additional solutions, hence we always have to check them against the initial conditions. Likewise, you should avoid dividing by 0, since that is rumored to cause the universe to implode.

\[ \boxed{ \begin{array} {l l} \text{ Then I'll huff, and I'll puff, and I'll blow your house in.} \\ \text{Not by the hair of my chinny chin chin!} \end{array} } \]

**3) State the techniques that you are using.**

You may have great familiarity with various techniques, and think that some statements are immediately obvious if you apply the XYZ principle. Your audience may not have the benefit of such an experience, and you can help them out by quickly illuminating the way.

\[ \boxed{ \text{ “Make a left turn 100 meters ahead.” - Siri} } \]

**4) Be clear about the given conditions.**

Remind the reader about the initial conditions when you are about to use them. This can help reduce confusion, especially if they didn't understand the conditions completely, thereby making it easier for them to piece the solution together. For algebra word problems, you should explain how you arrived at the mathematical statements, especially if you chose to add or multiply various terms.

\[ \boxed{ \mbox{ "Poof! Magic!" makes your proof go poof.} } \]

**5) Mark the end of your proof and state the conclusion.**

This indicates that you have completely answered the question that was stated, and the solution is finished.

Some members like to place their numerical answer in a box, which can be done using the < \boxed{ Answer } > command in Latex, which yields \( \boxed{123} \). To denote the end of a proof, some people write Q.E.D, which stands for Quod Erat Demonstrandum (Latin for "which was to be shown"). I prefer to use a small square, whose latex code is < _\square >, and which appears as \( _\square\).

\[ \boxed{ \text{ Q.E.D. = Quite Easily Done. } } \]

## Examples of solutions

Having outlined these guidelines, let's look at a few examples. The following solutions were written up for this question:

Do I look quadratic to you?:

The equation \( | x - 10 | = 7 \) is equivalent to the equation \( x^2 + ax + b = 0. \) What is the value of \( a+b \)?

Let’s look at the following solution:

Solution 1:We have \(x -10= 7 \) and so \( x = 17 \).

But then, we could also have \( x = 3 \).

So \( - 17 - 3 + 51 = 31 \).

How could the above guidelines help us to improve this?

**Guideline 1:**
The author likely meant that "Since \( |x-10| = 7 \), we either have \( x-10 = 7 \) or \( x - 10 = -7 \)". However, the way it is currently written, the second statement seems to appear magically from thin air.

**Guideline 2:** There is no logic between the first and second statement, which questions the validity of the entire argument. It is not clear why the second statement is true.

**Guideline 3:** If the author started off by stating the property of absolute values, he might realize that the first line isn’t completely correct.

**Guideline 4:** If the author reminded himself that the question stated \( | x-10| = 7 \), he would realize how to substantiate the first 2 statements.

**Guideline 5:** While this question may result in a correct answer of 31, it doesn’t explain all of the steps. What are the values of \(a\) and \(b\)? Why did he find the sum of 3 numbers when the question asked for the sum of 2 numbers?

Don’t forget to use the guidelines that you learnt as an Apprentice!

Now, let’s compare it to this solution:

Solution 2:By Daniel G.Absolute value of (x-10)=7 is also equal to (x-10)^2 = 49. Now the problem can be expanded so x^2-20x+100=49. The equation simplifies to x^2-20x+51.

So your a and b are -20 and 51, which gets you answer of31.

This solution provides the explanations for each of the steps, stating why the equations are equal to each other, and explaining that he expanded the terms to arrive at the algebraic mess, and then simplified it. Daniel started with a slight restatement of the given condition, and chose to bold the final answer, which can be done by typing \( {^*}{^*}31{^*}{^*} \). The mathematical statements are also easily understood and well explained, even without the use of LaTeX.

You can view this solution (and the problem) by clicking on the hyperlink Solution 2. If you enjoyed this solution, remember to vote it up!

Aspire to be better. Proceed on and be an Adept.

## Comments

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TopNewestHow can I post photo? – Chirag Gohil · 7 months ago

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When you click the image button in the formatting toolbar, you will be asked to select an image to upload. – Calvin Lin Staff · 7 months ago

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How to Include pictures in our solutions? – Debmeet Banerjee · 11 months, 2 weeks ago

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– Ashish Siva · 8 months, 1 week ago

If someone has already posted a question before, one cannot post another solution. We can only reply, if others another's to clarify. But while replying, we cannot include pictures. Is there another way to do it , sir? I do not edit the answers to the questions I have already posted.Log in to reply

For example, that is how I displayed the image in my above comment. – Calvin Lin Staff · 8 months, 1 week ago

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– Ashish Siva · 8 months, 1 week ago

Can we include videos in our answer. Like "IMO Problems Weekly"Log in to reply

Currently, we do not support video display on the site. – Calvin Lin Staff · 8 months, 1 week ago

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– Debmeet Banerjee · 11 months, 1 week ago

ThanksLog in to reply

can we post a solution only if we get the correct answer? – Saptarshi Sen · 9 months ago

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If you believe that the answer is incorrect, you can report the problem and explain why. If we update the answer to your value, then you will be able to add a solution afterwards. – Calvin Lin Staff · 9 months ago

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