Enter the coefficients and variables of your binomial expression to expand (a + b)² or (a − b)². Provide the first term coefficient, first term variable (optional), second term coefficient, and choose the sign (+ or −). The calculator returns the fully expanded trinomial, showing each step: the squared first term, the middle term, and the squared last term.
Expanded Expression
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First Term Squared (a²)
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Middle Term (±2ab)
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Last Term Squared (b²)
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The square of a binomial is the result of multiplying a two-term expression by itself. For example, (a + b)² means (a + b) × (a + b). Using the FOIL method or the binomial squared formula, this expands to a² + 2ab + b². The result is always a trinomial called a perfect square trinomial.
The rule states: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². In both cases, the expansion has three terms: the square of the first term, twice the product of both terms (positive for a sum, negative for a difference), and the square of the second term. This rule is one of the most fundamental identities in algebra.
To square a binomial difference such as (a − b)², apply the formula a² − 2ab + b². The key difference from the sum formula is that the middle term carries a negative sign. For example, (6 − b)² = 36 − 12b + b². Never confuse this with a² − b², which is the difference of two squares, not a squared binomial.
When you square a binomial, you always get a perfect square trinomial — a polynomial with exactly three terms. The first and last terms are always positive (since squaring any real number yields a non-negative result), while the middle term may be positive or negative depending on the sign of the original binomial.
Perfect square trinomials are polynomials of the form a² + 2ab + b² or a² − 2ab + b² that can be factored back into (a + b)² or (a − b)² respectively. Recognizing them is useful in factoring, completing the square, and solving quadratic equations. Any trinomial where the first and last terms are perfect squares and the middle term equals twice the product of their square roots is a perfect square trinomial.
If one term in the binomial is 0, then the expansion simplifies considerably. For example, (a + 0)² = a² + 0 + 0 = a², giving just the square of the remaining term. This makes sense because squaring a monomial only squares that single term with no cross-product or constant contribution.
Yes. The formula works for any algebraic expressions in the positions of a and b. For instance, (3x + 4y)² = (3x)² + 2(3x)(4y) + (4y)² = 9x² + 24xy + 16y². The same logic applies regardless of whether the terms include variables, coefficients, or even more complex sub-expressions.
When you expand (a + b)(a + b) using the FOIL method, the outer product gives 'ab' and the inner product also gives 'ab'. These two like terms combine to produce 2ab. This doubling is why the binomial squared formula always has a coefficient of 2 in the middle term — it reflects the two identical cross-products generated during multiplication.