Question

Determine the equivalent expression of $(a-b{)}^2$.

• A. $a^2+2ab-b^2$
• B. $a^2-2ab-b^2$
• C. $a^2+2ab+b^2$
• D. $a^2-2ab+b^2$

Answer 1

The correct option is D $a^2-2ab+b^2$

Geometrically, if a square is divided in four different geometrical shapes, the area of the square will be the sum of the areas of the four individual geometrical shapes. It can be written in mathematical form as an equation.

$a^2=(a-b{)}^2+b(a-b)+(a-b)b+b^2$

In this derivation, we have to find the expansion of $(a-b{)}^2$ identity. So, shift all the terms to other side of the equation for finding the equivalent value of the $(a-b{)}^2$.

$\Rightarrow a^2-b(a-b)-(a-b)b-b^2=(a-b{)}^2$

$\Rightarrow (a-b{)}^2=a^2-b(a-b)-(a-b)b-b^2$

In the right hand side of the equation, the second and third terms $b(a-b)$ and $(a-b)b$ are mathematically equal as per the commutative property of multiplication.

$(a-b{)}^2=a^2-b(a-b)-b(a-b)-b^2$

$\Rightarrow (a-b{)}^2=a^2-2b(a-b)-b^2$

$\Rightarrow (a-b{)}^2=a^2-2ba+2b^2-b^2$

$\therefore (a-b{)}^2=a^2-2ab+b^2$

It can also be written as $(a-b{)}^2=a^2+b^2-2ab$.