Solved Examples

Example 1: Mind the Mistake

Incorrect Expansion: $(a + 3)^2 = a^2 + 9$ Explanation: This is a common error where the middle term is forgotten. Correct: $(a + 3)^2 = a^2 + 2(a)(3) + 3^2 = a^2 + 6a + 9$

Example 2: Negative Distributivity

Expand: $-3x(2y - 5)$ $= (-3x)(2y) + (-3x)(-5)$ $= -6xy + 15x$

Example 3: Fractions

Simplify: $(x + \frac{1}{x})^2$ Using Identity 1A: $x^2 + 2(x)(\frac{1}{x}) + (\frac{1}{x})^2$ $= x^2 + 2 + \frac{1}{x^2}$

Example 4: Large Numbers

Calculate: $53 \times 47$ Recognize this as $(50+3)(50-3)$. $50^2 - 3^2$ $= 2500 - 9 = 2491$