Understanding the Square of a Binomial
The square of a binomial is a special product in algebra. A binomial is an expression with two terms (like a + b or a - b). Squaring it means multiplying the binomial by itself.
The Two Formulas
Sum Squared
When the binomial has a plus sign between the terms.
(a + b)² = a² + 2ab + b²
Difference Squared
When the binomial has a minus sign between the terms.
(a - b)² = a² - 2ab + b²
Derivation by FOIL Method
(a + b)² Derivation
(a + b)² = (a + b)(a + b)
- First: a x a = a²
- Outer: a x b = ab
- Inner: b x a = ab
- Last: b x b = b²
Combine: a² + ab + ab + b² = a² + 2ab + b²
(a - b)² Derivation
(a - b)² = (a - b)(a - b)
- First: a x a = a²
- Outer: a x (-b) = -ab
- Inner: (-b) x a = -ab
- Last: (-b) x (-b) = b²
Combine: a² - ab - ab + b² = a² - 2ab + b²
Common Mistakes to Avoid
- Wrong: (a + b)² = a² + b² -- this is incorrect because it omits the middle term 2ab.
- Right: (a + b)² = a² + 2ab + b² -- always include the middle term.
- For (a - b)², the middle term is negative (-2ab), but the last term (b²) is always positive.
Applications
The square of a binomial is widely used in completing the square, deriving the quadratic formula, simplifying algebraic expressions, and in many areas of mathematics, physics, and engineering.