Completion (algebra)
Completion in algebra is like putting together a puzzle. Imagine you have a puzzle but there are a few missing pieces. You can't finish the picture without those missing pieces. But what if you had some extra pieces that could fit in those missing spots? That's what completion is like.
Let's say you have an equation like this: x^2 - 5x + 6 = 0. You want to find the solutions for x, but you can't just solve it with regular algebra because it's not in a nice, simple form. But what if you added a new number, let's call it "y", to the equation so that it becomes a perfect square?
To do this, we add and subtract y to the equation like this: x^2 - 5x + y^2 - y^2 + 6 = 0. Now we can rewrite it as (x - 5/2)^2 = y^2 - 1/4.
This looks more complicated but it's actually simpler because now we can easily solve for x. We know that (x - 5/2)^2 is equal to something squared minus 1/4, so it must be the difference of two squares.
Therefore, we can write it as (x - 5/2 + y/2)(x - 5/2 - y/2) = 0. This means that either (x - 5/2 + y/2) or (x - 5/2 - y/2) must equal 0.
We can solve for x in each case and get two solutions: x = 5/2 - y/2 or x = 5/2 + y/2. These are the solutions we were looking for.
So completion is like adding missing puzzle pieces to an equation so that it becomes a perfect square, which makes it easier to solve.
Let's say you have an equation like this: x^2 - 5x + 6 = 0. You want to find the solutions for x, but you can't just solve it with regular algebra because it's not in a nice, simple form. But what if you added a new number, let's call it "y", to the equation so that it becomes a perfect square?
To do this, we add and subtract y to the equation like this: x^2 - 5x + y^2 - y^2 + 6 = 0. Now we can rewrite it as (x - 5/2)^2 = y^2 - 1/4.
This looks more complicated but it's actually simpler because now we can easily solve for x. We know that (x - 5/2)^2 is equal to something squared minus 1/4, so it must be the difference of two squares.
Therefore, we can write it as (x - 5/2 + y/2)(x - 5/2 - y/2) = 0. This means that either (x - 5/2 + y/2) or (x - 5/2 - y/2) must equal 0.
We can solve for x in each case and get two solutions: x = 5/2 - y/2 or x = 5/2 + y/2. These are the solutions we were looking for.
So completion is like adding missing puzzle pieces to an equation so that it becomes a perfect square, which makes it easier to solve.
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