The Square of a Binomial Calculator provided by Hesapstan expands (a+b)² and (a−b)² for numeric a and b values. It shows a², the middle term ±2ab, b², and the combined result so the most common mistake, treating (a+b)² as a²+b², is easy to catch.
The square of a binomial always includes a middle term
A square of a binomial is the square of a sum or difference of two terms. The identities are (a+b)²=a²+2ab+b² and (a−b)²=a²−2ab+b². This calculator makes each part visible instead of jumping straight to the final number.
(a+b)² is not equal to a²+b². Squaring the binomial also multiplies the two terms together, and that interaction creates the middle term 2ab.
The calculator has separate plus and minus modes
Choose the plus mode for (a+b)² and the minus mode for (a−b)². In both cases, a and b are numeric values, and the calculator applies the corresponding identity step by step.
- In (a+b)² mode, the terms are a², +2ab, and b².
- In (a−b)² mode, the terms are a², −2ab, and b².
- The output shows the identity line, the substituted expression, each term, and the combined result.
Here a and b are raw numbers. Expressions such as (px+q)² are polynomial expressions and belong in a different calculator, such as a perfect-square trinomial or polynomial expansion tool.
A plus example shows why a²+b² is incomplete
For a=3 and b=4, the expression (3+4)² must be expanded as a²+2ab+b². This example shows exactly why 3²+4² is not enough.
- First term: a² = 3² = 9.
- Middle term: +2ab = 2×3×4 = 24.
- Last term: b² = 4² = 16.
- Final result: 9+24+16 = 49, matching (3+4)² = 7² = 49.
3²+4²=25 only adds the first and last terms. The correct value of (3+4)² is 49 because the middle term 2ab equals 24.
A minus example changes the sign of the middle term
In minus mode, the structure is the same but the middle term is negative. For a=5 and b=2, (5−2)² expands as 5²−2×5×2+2².
- First term: a² = 5² = 25.
- Middle term: −2ab = −2×5×2 = −20.
- Last term: b² = 2² = 4.
- Final result: 25−20+4 = 9, matching (5−2)² = 3² = 9.
In (a−b)² mode, if a and b are equal, the result is 0. The middle term still appears; for example, (4−4)² gives 16−32+16=0.
Decimals and negative numbers are supported, with numeric output
The calculator supports integers, decimals, zero, and negative values for a and b. For example, if a=1.5 and b=0.5, then (a+b)² becomes 1.5²+2×1.5×0.5+0.5² = 2.25+1.5+0.25 = 4.
Negative values follow the normal rules of multiplication. For example, a=-3 and b=2 in plus mode gives (-3+2)² = (-1)² = 1.
Decimal arithmetic is displayed with trimmed formatting. Very small floating-point artifacts may be rounded away by the calculator display.
This is related to FOIL, but it is not the full FOIL calculator
A binomial square is a special case of multiplying two binomials: (a+b)(a+b) or (a−b)(a−b). This calculator focuses on the square identity and the middle term, not on the full First-Outer-Inner-Last sequence.
- FOIL expands general products such as (ax+b)(cx+d).
- A perfect-square trinomial tool focuses on recognizing polynomial forms such as x²+6x+9.
- This calculator expands numeric (a+b)² or (a−b)² only.
- Reverse FOIL or box-method tools are better for factoring trinomials.
The calculator does not solve equations or produce symbolic polynomial output
This tool substitutes numeric a and b values into the selected identity. It does not solve equations such as (a+b)²=c, find roots, or produce symbolic polynomial expansions with variables.
If you need to expand something like (2x+3)², use the FOIL calculator, a perfect-square trinomial tool, or a polynomial expansion tool instead.
Frequently Asked Questions
Why is (a+b)² not equal to a²+b²?
Because (a+b)² means (a+b)(a+b). When you multiply the binomial by itself, two ab terms appear, and together they form the middle term 2ab.
Can I enter negative values?
Yes. a and b can be negative. The calculator evaluates a², ±2ab, and b² using the normal sign rules.
Are decimals supported?
Yes. Decimal values such as 1.5 and 0.5 are supported. The result is numeric and may be formatted to remove unnecessary trailing zeros.
What is the difference between this calculator and a perfect-square trinomial tool?
This calculator uses numeric a and b values to evaluate (a±b)². A perfect-square trinomial tool works with polynomial forms such as x²+6x+9.
When should I use the FOIL calculator instead?
Use FOIL when the factors are two general binomials, such as (2x+3)(4x−5). Use this calculator when the expression is specifically the square of a sum or difference.