Algebraic Expressions and Identities - Applying Identities

An Algebraic Identity is an equality that holds true for every value of the variable(s) present in it. Think of it like a balance scale where both sides are identical in value regardless of the weights (numbers) you substitute into the letters.

The identity $(a + b)^2 = a^2 + 2ab + b^2$ represents the area of a large square with side length $(a+b)$. If you visualize this square divided into four sections based on lengths $a$ and $b$, you will see one square of area $a^2$, another square of area $b^2$, and two identical rectangles each with area $ab$.

The identity $(a - b)^2 = a^2 - 2ab + b^2$ is used for the square of a binomial difference. Visually, this can be understood by starting with a square of side $a$ and removing two strips of width $b$, then adding back the small square $b^2$ that was subtracted twice at the corner.

The identity $(a + b)(a - b) = a^2 - b^2$ is known as the difference of two squares. Geometrically, if you have a square of area $a^2$ and remove a smaller square of area $b^2$ from the corner, the remaining L-shaped area can be rearranged into a rectangle with dimensions $(a+b)$ and $(a-b)$.

The identity $(x + a)(x + b) = x^2 + (a + b)x + ab$ applies when the first terms of two binomials are identical. Imagine a rectangle with width $(x + a)$ and height $(x + b)$; its total area is composed of a square $x^2$, two rectangles $ax$ and $bx$, and a smaller rectangle $ab$.

Identities are powerful tools for mental math. For example, to calculate $103^2$, you can visualize it as $(100 + 3)^2$ and apply the square of a sum identity to avoid tedious long multiplication.

When applying identities, the first step is always identifying the values of $a$ and $b$ (or $x$) by comparing the given expression with the standard form of the identity. For example, in $(3x - 5y)^2$, $a = 3x$ and $b = 5y$.