Square of a Binomial Calculator — (a + b)² Expansion
Enter values for a and b to instantly expand (a + b)² into a² + 2ab + b². Negative values of b give the (a − b)² identity automatically. Every step is shown so you can follow the algebra.
Square of the binomial (a + b)
Write the identity
Substitute a and b
Compute a²
Compute 2ab
Compute b²
Sum all three terms
- 1
a²
3² = 9 - 2
2ab
2 × 3 × 4 = 24 - 3
b²
4² = 16 - 4
(a + b)²
9 + 24 + 16 = 49The middle term 2ab is the cross term that appears when a binomial is squared.
How does this calculator work?
The square of a binomial follows (a + b)² = a² + 2ab + b². Enter a and b to see each term computed (a², 2ab, b²) and summed, with a full step-by-step expansion. Use a negative b to get the (a − b)² form. The middle term 2ab is the key that beginners miss.
Formula
How this is calculated
The square of a binomial is a fundamental algebraic identity. Multiplying (a + b) by itself — (a + b)(a + b) — distributes into four products: a·a + a·b + b·a + b·b, which simplifies to a² + 2ab + b². The middle term 2ab (the "cross term") is what beginners often overlook when squaring a sum by hand.
When b is negative the same identity gives the (a − b)² form: substitute −b for b and you get a² − 2ab + b². The calculator handles this automatically — just enter b as a negative number.
This identity underlies completing the square (solving quadratics), expanding polynomial expressions, and deriving Pythagorean-type results. It assumes real numbers; for complex numbers, the expansion holds algebraically but the interpretation of "squared" changes.
Frequently asked questions
(a + b)² = a² + 2ab + b². It says that squaring a two-term sum equals the square of the first term, plus twice the product of both terms, plus the square of the second term. For (a − b)², substitute a negative b to get a² − 2ab + b².
When you expand (a + b)(a + b) using FOIL, the outer product is a·b and the inner product is also b·a — both equal ab. Adding them gives 2ab. This cross term is the most common mistake when squaring a binomial by hand.
A perfect square trinomial is any expression of the form a² + 2ab + b² (or a² − 2ab + b²) because it is the exact square of a binomial. Recognising this pattern lets you factor or simplify expressions quickly.
Also known as
TG we-Calculate Editorial Team. (2026). Square of a Binomial Calculator — (a + b)² Expansion [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/square-of-a-binomial-calculator
TG we-Calculate Editorial Team. "Square of a Binomial Calculator — (a + b)² Expansion." TG we-Calculate. 2026. https://we-calculate.com/calculator/square-of-a-binomial-calculator.
TG we-Calculate Editorial Team, "Square of a Binomial Calculator — (a + b)² Expansion," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/square-of-a-binomial-calculator
@misc{wecalculate_square_of_a_binomial_calculator, title = {Square of a Binomial Calculator — (a + b)² Expansion}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/square-of-a-binomial-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?