Algebraic Expressions and Identities - Standard Identities: (a+b)², (a-b)², a²-b², (x+a)(x+b)

An algebraic identity is an equality that holds true regardless of the values assigned to its variables. Unlike a standard equation which is true only for specific values of $x$, an identity like $(a+b)^2 = a^2 + 2ab + b^2$ is a universal rule used to simplify expressions and perform mental calculations quickly.

The identity $(a+b)^2 = a^2 + 2ab + b^2$ can be visualized as the total area of a large square with side length $(a+b)$. If you divide this large square into four sections, you get one square of area $a^2$, another square of area $b^2$, and two identical rectangles each having an area of $ab$. Adding these four areas together gives the complete formula.

The identity $(a-b)^2 = a^2 - 2ab + b^2$ represents the area of a smaller square with side length $(a-b)$. Visually, if you start with a large square of area $a^2$ and remove two rectangles of area $ab$, you have removed the corner square $b^2$ twice. To correct this, we add $b^2$ back once, resulting in the final expression.

The identity $a^2 - b^2 = (a+b)(a-b)$ is known as the Difference of Two Squares. Imagine a large square of side $a$ with a smaller square of side $b$ cut out from its corner. The remaining L-shaped region can be sliced and rearranged into a single rectangle with dimensions $(a+b)$ and $(a-b)$, proving that their areas are identical.

The identity $(x+a)(x+b) = x^2 + (a+b)x + ab$ is used when multiplying two binomials that share a common first term $x$. Visually, this is a rectangle with length $(x+a)$ and width $(x+b)$. The area is composed of a square $x^2$, two rectangles with areas $ax$ and $bx$ (which combine to $(a+b)x$), and a small rectangle with area $ab$.

Identities are powerful tools for mental arithmetic. For example, to find $105^2$, we can treat it as $(100+5)^2$ and apply the first identity. Similarly, products like $98 \times 102$ can be solved as $(100-2)(100+2)$ using the third identity, making complex multiplication much simpler.