What is (a+b)²

Understand the Problem

The question asks for the expansion of the binomial expression (a+b)². This is a fundamental algebraic identity. We need to expand the expression using the distributive property or by recalling the standard formula.

Answer

$a^2 + 2ab + b^2$

Steps to Solve

Recognize the binomial square

We have to expand the square of a binomial $(a+b)^2$.

Apply the distributive property (FOIL method)

$(a+b)^2$ can be rewritten as $(a+b)(a+b)$. Now, we apply the distributive property (also known as the FOIL method):

$(a+b)(a+b) = a(a+b) + b(a+b)$

Distribute again

Now distribute $a$ and $b$ in the respective terms:

$a(a+b) + b(a+b) = a^2 + ab + ba + b^2$

Simplify

Since $ab = ba$, we can combine these terms:

$a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$

$a^2 + 2ab + b^2$

More Information

The binomial expansion of $(a+b)^2$ is a fundamental algebraic identity. It appears frequently in various mathematical contexts, from basic algebra to calculus and beyond. It's worth memorizing!

Tips

A common mistake is to incorrectly expand $(a+b)^2$ as $a^2 + b^2$. The $2ab$ term is often forgotten. Remember to expand $(a+b)^2$ as $(a+b)(a+b)$ and carefully apply the distributive property to avoid this error.

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