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Timothy Gan

(a + b)² = a² + b² + **2ab**

- Why (a + b)² ≠ a² + b² ?

This is probably one of the most common mathematics mistakes we have done at least once in our life, which is **(a+b)² = a²+b²**. Now what we need to understand is why is this not the case. Now before we start, we just have to understand the idea of a square. The concept of a square is pretty simple. Say if we want **a²**, **a²** just means that we have a square of sides **a**. Thus, the area is multiplied by **a**. So, similarly for **b²**.

What we need to understand is the length of this side here is **b** and this side here is **b** and hence the area here is **b²**. So what does **(a+b)²** mean? This would mean that we have a huge square. Now in this case here, we're going to split one side into **a** and **b** and this side into **a** and **b**. This means that we have a side of **a** and this side will be **b**.

The whole side will just be **a+b**. And this will give us what we are looking for, which is an **(a+b)²**. So in this case here, if you look at the big square here, we know that if we expand in this form [**(a + b)² = a² + b²**] we're actually missing out on something (Hint: purple colour in the diagram), right?

The area which we did not account for when we wrote such a common mistake is actually **ab-ab** right? So in other words, to fix the expansion, we need to just put **2ab** in to get the correct expansion: