Hey, this is Presh Talwalkar. Here's a portion of an email I received.
Hi I'm Lucas from Brazil, and I love your videos.
Here's a problem from a book with math problems that I could not solve and the book doesn't say the answer,
it just said "This one we leave to the students!"
I translated the problem for you, and i'm also sending you a photo of the original problem
and the book link from where I got the problem
I was so grateful to Lucas, I replied right away. "Thanks for writing and for translating too, I appreciate the original photo"
So what was the problem? It was a false proof that 3 is equal to 0
Here's how it starts.
Let x be a solution of x squared plus x plus 1 equals 0
Since x is not equal to 0 we can divide both sides by x
Let's simplify this to the following form we have x plus 1, plus 1 over x is equal to 0
Now, from the first equation we have x plus 1 is equal to negative x squared
We're going to substitute that into the second equation
so that first x plus 1 becomes negative x squared, then we have plus 1 over x is equal to 0
We'll now rearrange this equation to get 1 over x is equal to x squared
This means 1 is equal to x cubed so x is equal to 1
We now take x equals 1 and
substitute it into our original equation x squared plus x plus 1 equals 0
We get 1 squared plus 1 plus 1 is equal to 0 which means 3 is equal to 0
This is clearly a nonsensical result
But every single step seemed to be correct
So where is the mistake in this false proof that 3 is equal to 0?
Can you figure it out?
Give this problem a try and when you're ready keep watching the video for the explanation.
So let's take a look at this false proof
Which step introduces the mistake from which the ultimate result of three equals zero originates?
That would be between the second and third steps where we substitute x plus 1 is equal to negative x squared
The problem with this step is that it creates an extraneous
solution x equals 1, which is not a solution to the original equation, x squared
plus x plus 1 equals 0
So let me explain that in a little bit more detail
We start out where x is a solution of x squared plus x plus 1 equals 0, so far so good
This has two solutions, which we can get from the quadratic formula
We have negative 1 plus i times the square root of 3 all over 2, and we have negative 1 minus
i times the square root of 3 all over 2
Now x is not equal to 0 so it is valid for us to divide both sides by x
Now we have a new equation x plus 1 plus 1 over x is equal to 0
This equation again has the same two solutions
So far so good
Now, we're going to substitute x plus 1 is equal to negative x squared
We get the equation negative x squared plus 1 over x is equal to 0. Now what happens when you solve this equation?
Well you end up with another solution so you have the same two solutions as before
but then you end up with the new solution of x equals 1
Notice if you substitute x equals 1 into this equation you end up with negative 1 plus 1 equals 0
So x equals 1 is a solution to this equation
But it was not a solution to the previous two equations
So the lesson is that when we've substituted x plus 1 is equal to negative x squared,
we've got an equation that has another solution,
and this is not a solution to the original equation x squared plus x plus 1 is equal to 0
So you have to be careful if a step introduces an extraneous solution,
Or else you could end up with a nonsensical result like three is equal to zero
Did you identify
where the mistake originated?
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I make videos on math you can catch me on my blog
Mind Your Decisions. If you like this video you can check out my books which are linked in the video description and you can support
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