The area of a rectangle is $(x^2-49)$ sq cm and its length is $(x+7)$ cm. What is its breadth?

Read the identity backwards: $x^2-49=x^2-7^2=(x+7)(x-7)$. Since one factor is the given length $(x+7)$, the breadth is $(x-7)$ cm. Check: $(x+7)(x-7)=x^2-49$. Area is a PRODUCT of the sides, so it must be factorised — subtracting the length from the area gives $x^2-x-56$, and splitting the terms one by one gives $x-49$.