Binomial Formulas

Calculator for expanding and evaluating the three binomial formulas

Binomial Formulas Calculator

Binomial Formulas

Computes the expansion of binomial expressions with two variables a and b by systematic multiplication of the three standard formulas

What are binomial formulas?

Binomial formulas are algebraic identities for expanding products of binomials (expressions with two terms). They simplify calculations and help avoid mistakes.

Select a binomial formula
First formula
\((a + b)^2\)
\(a^2 + 2ab + b^2\)
Second formula
\((a - b)^2\)
\(a^2 - 2ab + b^2\)
Third formula
\((a + b)(a - b)\)
\(a^2 - b^2\)
Variable a
Numeric value
Variable b
Numeric value
Binomial formula result
Select a formula and enter values
The binomial formula will be computed using the entered values

Formula Info

Binomial formulas

Memory formulas: Simplify expansion of binomials

(a±b)² a²±2ab+b² Expansion

First: (a+b)² = a²+2ab+b²
Second: (a-b)² = a²-2ab+b²
Third: (a+b)(a-b) = a²-b²

Examples
(3+2)²: 9 + 12 + 4 = 25
(5-1)²: 25 - 10 + 1 = 16
(4+3)(4-3): 16 - 9 = 7
Memory trick
First formula: "Plus"
Second formula: "Minus"
Third formula: "Difference of squares"

The three binomial formulas

First binomial formula
\[(a + b)^2 = a^2 + 2ab + b^2\]
Derivation: \[(a + b)^2 = (a + b)(a + b)\] \[= a^2 + ab + ba + b^2 = a^2 + 2ab + b^2\]
Second binomial formula
\[(a - b)^2 = a^2 - 2ab + b^2\]
Derivation: \[(a - b)^2 = (a - b)(a - b)\] \[= a^2 - ab - ba + b^2 = a^2 - 2ab + b^2\]
Third binomial formula
\[(a + b)(a - b) = a^2 - b^2\]
Derivation: \[(a + b)(a - b)\] \[= a^2 - ab + ba - b^2 = a^2 - b^2\]

Calculation examples for binomial formulas

Example 1: (7 + 3)²
a = 7, b = 3
\[\begin{aligned} (7 + 3)^2 &= 7^2 + 2 \cdot 7 \cdot 3 + 3^2 \\ &= 49 + 42 + 9 \\ &= 100 \end{aligned}\]

First binomial formula

Example 2: (8 - 3)²
a = 8, b = 3
\[\begin{aligned} (8 - 3)^2 &= 8^2 - 2 \cdot 8 \cdot 3 + 3^2 \\ &= 64 - 48 + 9 \\ &= 25 \end{aligned}\]

Second binomial formula

Example 3: (6 + 4)(6 - 4)
a = 6, b = 4
\[\begin{aligned} (6 + 4)(6 - 4) &= 6^2 - 4^2 \\ &= 36 - 16 \\ &= 20 \end{aligned}\]

Third binomial formula

Recognition rules
First formula

• Both terms added

• Squared: (a+b)²

• Middle term: +2ab

Second formula

• Terms subtracted

• Squared: (a-b)²

• Middle term: -2ab

Third formula

• Sum × difference

• (a+b)(a-b)

• Only: a² - b²

The middle terms (±2ab) vanish in the third formula!

Applications of binomial formulas

Binomial formulas are fundamental algebra tools with broad applications:

School mathematics
  • Term transformations and simplifications
  • Solving equations and factoring
  • Mental arithmetic for special numbers
  • Geometric area calculations
Higher mathematics
  • Polynomial division and factorization
  • Simplifying integrals and derivatives
  • Complex numbers and algebraic structures
  • Binomial theorem and combinatorics
Natural sciences
  • Physics: energy and momentum conservation
  • Chemistry: reaction equations
  • Engineering: optimization problems
  • Statistics: variance calculations
Practical applications
  • Simplifying compound interest calculations
  • Geometric constructions
  • Computer graphics: coordinate transformations
  • Signal processing and Fourier analysis

Binomial formulas: Foundation of algebra

The binomial formulas are among the most fundamental tools of algebra and form the basis for more advanced mathematical concepts. These three elegant identities - (a±b)² = a² ± 2ab + b² and (a+b)(a-b) = a² - b² - transform seemingly complicated multiplications into manageable standard forms. From elementary term manipulation to higher analysis, from geometric area calculations to complex numbers, these formulas permeate mathematics as indispensable computational tools and structural building blocks.

Summary

The binomial formulas combine algebraic elegance with practical applicability. Their apparent simplicity hides deep mathematical structures that span from basic arithmetic to modern algebra. As mnemonic formulas they ease calculations and prevent errors; as mathematical identities they open new solution paths and connections. From efficient computation of special products to factoring complex terms and geometric interpretation of areas, binomial formulas show how fundamental mathematical principles solve practical problems elegantly and systematically.

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