Question

Factor the expression completely.

${\left(a+b\right)}^2-{\left(a-b\right)}^2$

Short answer

The solution${\left(a+b\right)}^2-{\left(a-b\right)}^2=4ab$

Step-by-step solution

Step 1. Apply the “square of the difference” formula.

If A and B are any real numbers or algebraic expressions, then

${\left(A-B\right)}^2=A^2-2AB+B^2$

Step 2. Apply the “square of the sum” formula.

If A and B are any real numbers or algebraic expressions, then

${\left(A+B\right)}^2=A^2+2AB+B^2$

Step 3. Analyze the given expression.

Given${\left(a+b\right)}^2-{\left(a-b\right)}^2$

We will expand both the expressions first to get:

${\left(a+b\right)}^2-{\left(a-b\right)}^2=\left[a^2+2ab+b^2\right]-\left[a^2-2ab+b^2\right]$

Step 4. Solve the expression.

Simplify the expression:

$\begin{array}{l}{\left(a+b\right)}^2-{\left(a-b\right)}^2=\left[a^2+2ab+b^2\right]-\left[a^2-2ab+b^2\right]\\ \Rightarrow {\left(a+b\right)}^2-{\left(a-b\right)}^2=a^2+2ab+b^2-a^2+2ab-b^2\\ \Rightarrow {\left(a+b\right)}^2-{\left(a-b\right)}^2=a^2-a^2+2ab+2ab+b^2-b^2\\ \Rightarrow {\left(a+b\right)}^2-{\left(a-b\right)}^2=4ab\end{array}$