Algebraic Expressions and Identities - Application of Identities

Definition of an Identity: An algebraic identity is an equality that remains true for all possible values of the variables involved. Unlike a standard equation which is only true for specific values, an identity represents a universal rule for transforming expressions.

Identity I (Square of a Sum): The expression $(a + b)^2 = a^2 + 2ab + b^2$ is the expansion of a binomial square. Visually, imagine a large square with a side length of $(a + b)$. The total area is composed of one square with area $a^2$, another square with area $b^2$, and two identical rectangles each with area $ab$, filling the total space of the $(a+b)$ by $(a+b)$ square.

Identity II (Square of a Difference): The expression $(a - b)^2 = a^2 - 2ab + b^2$ handles the subtraction of terms. Geometrically, this represents the area of a smaller square (with side $a - b$) that remains when you start with a square of side $a$ and subtract the regions representing $2ab$ while adding back the $b^2$ that was subtracted twice.

Identity III (Difference of Two Squares): The product $(a + b)(a - b) = a^2 - b^2$ is a fundamental tool for factorization. Visually, if you take a square of area $a^2$ and cut out a smaller square of area $b^2$ from the corner, the remaining L-shaped region can be sliced and rearranged into a single rectangle with dimensions $(a + b)$ and $(a - b)$.

Identity IV (Product of Binomials with a Common Term): The identity $(x + a)(x + b) = x^2 + (a + b)x + ab$ is used when the first terms of two binomials are identical. The result is a trinomial where the middle term's coefficient is the sum of the non-common terms and the final constant is their product.

Numerical Applications: Identities can be used to perform complex arithmetic mentally. For example, calculating $102 \times 98$ can be simplified using the Difference of Squares identity by rewriting it as $(100 + 2)(100 - 2)$, which equals $100^2 - 2^2$.

Simplification and Evaluation: Standard identities are often used to simplify long algebraic strings or to find the value of an expression like $x^2 + \frac{1}{x^2}$ given the value of $x + \frac{1}{x}$ by squaring both sides of the given equation.