Question: Cambridge Dictionary gives this definition about "complain"

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> to say that something is wrong or not satisfactory
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and this definition about "complaint"

> 
> a statement that something is wrong or not satisfactory
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I guess both of them means the same thing. However, [Google Ngram](https://books.google.com/ngrams/graph?content=to%20complain%20about%2C%20make%20a%20complaint%20about%20&case_insensitive=on&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cto%20complain%20about%3B%2Cc0%3B.t1%3B%2Cmake%20a%20complaint%20about%3B%2Cc0#t1%3B%2Cto%20complain%20about%3B%2Cc0%3B.t1%3B%2Cmake%20a%20complaint%20about%3B%2Cc0) shows that the former is much more commonly used that the latter

[![enter image description here](https://i.stack.imgur.com/IyIrh.png)](https://i.stack.imgur.com/IyIrh.png)

Are those two expressions interchangeable?

Answer: 
When in doubt, I find it helpful to simplify the sentence. Consider these: 

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> The conclusion is uncertain.  
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>  The conclusion is final.  
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>  The conclusion is X.
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Clearly, whatever the conclusion is, it's singular and needs a singular verb. 

Now let's look at what the actual conclusion is:

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> Both are harmful.
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Again, clearly, "both" refers to two things and thus requires a plural verb, "are". You could check that by replacing both:

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> Cigarettes and gun battles are harmful.  
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>  Angry dragons and mean dogs are harmful.
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> 

Now, let's take that last *conclusion* example -- *The conclusion is X.* -- and replace X with what it actually is:

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> The conclusion is [both are harmful]. 
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> 

When you look at it that way, it becomes clear that the sentence as originally written is indeed correct.

My conclusion **is** that your friends **are** wrong.
You have two ideas together. One is that the conclusion *is* right. The other is that both A and B *are* harmful. You are putting these into one sentence and finding it troublesome. 

You are also rushing by with spoken English and leaving out an implied word that explains things.

The conclusion is *that* both are purple.