Question

Evaluate the following equation
 \[{a^2} - {b^2} - {({a^{}} + b)^2}\]

Answer 1

Hint: Here, we will be proceeding by firstly opening all the square brackets given in the equation. For opening the brackets, we need to know the basic identities. After opening all the brackets, we will have to do the simple mathematics. After this, we will get our required answer.

Complete step-by-step answer:
Given, ${a^2} - {b^2} - {(a + b)^2}$
Firstly, we need to open all the squares and brackets.
After opening the squares, the equation becomes
${a^2} - {b^2} - ({a^2} + 2ab + {b^2})$
Because ${(a + b)^2} = {a^2} + 2ab + {b^2}$
And after opening the brackets, the equation becomes
${a^2} - {b^2} - {a^2} - 2ab - {b^2}$
Now, it is the time to do the simple mathematics
After doing simple addition, the equation becomes
$ - {b^2} - 2ab - {b^2}$
After simplifying, the equation becomes
$ \Rightarrow - 2{{\text{b}}^2} - 2ab = - 2b(b + a)$
The above equation can also be written as
$ - 2b(a + b)$
It is our required answer.

Note: In these types of problems, firstly we need to simplify our given equation. For simplifying, we need to open all the squares and brackets. Like in the given equation, firstly we opened the square and the bracket to make the equation simpler to solve. The main thing to solve these types of equations is to reduce the complexity of the equation step by step.