Square of a Binomial Calculator – Guide & Formulas
Expand binomial squares instantly. Enter a and b to compute (a + b)² and (a - b)² with step-by-step expansion and verification.
Expand any binomial square instantly with our free calculator. Enter a and b to compute both (a + b)² and (a - b)² with step-by-step expansion and verification.
Key Takeaway
Use the free Square of a Binomial Calculator to expand binomial squares instantly. enter a and b to compute (a + b)² and (a - b)² with step-by-step expansion and verification. Get instant results with step-by-step explanations.
How to Use the Square of a Binomial Calculator
- Enter the value or expression for a (first term of the binomial).
- Enter the value or expression for b (second term of the binomial).
- Review the expanded forms for both (a + b)² and (a - b)².
- Check the step-by-step FOIL expansion and verification.
The Formula
Variable Definitions
- a: The first term of the binomial
- b: The second term of the binomial
- 2ab: The middle (cross) term — positive for sum, negative for difference
- a²: The square of the first term
- b²: The square of the second term
Expanding (3x + 4)²
Apply the binomial square formula to a linear expression.
- Step 1: Identify a = 3x and b = 4.
- Step 2: Apply (a + b)² = a² + 2ab + b².
- Step 3: Compute each term: a² = (3x)² = 9x², 2ab = 2(3x)(4) = 24x, b² = 4² = 16.
- Step 4: Combine: (3x + 4)² = 9x² + 24x + 16.
Frequently Asked Questions
What is the square of a binomial?
The square of a binomial (a + b)² expands to a² + 2ab + b². The square of (a - b)² expands to a² - 2ab + b². This is a fundamental algebraic identity used throughout mathematics.
Why is the middle term 2ab and not ab?
When you multiply (a + b)(a + b) using FOIL, the outer and inner products both give ab, so you get ab + ab = 2ab. This is why the middle term is always doubled.
What is the common mistake with binomial squares?
The most common error is writing (a + b)² = a² + b², forgetting the middle term 2ab. Always apply the full formula: a² + 2ab + b².
Can I use this for expressions with variables?
Yes. For example, (2x + 3y)² = 4x² + 12xy + 9y². Treat each term as a single unit and apply the formula.
How does this relate to the FOIL method?
Squaring a binomial is a special case of FOIL: (a + b)(a + b) = First + Outer + Inner + Last = a² + ab + ab + b² = a² + 2ab + b².
What about (a + b)³?
The cube uses the binomial theorem: (a + b)³ = a³ + 3a²b + 3ab² + b³. This calculator focuses on the square (second power) only.
Can a or b be negative?
Yes. For (a + b)², if b is negative, say b = -c, then (a - c)² = a² - 2ac + c². The formula handles signs automatically.
How do I verify my expansion?
Multiply the original binomial by itself: (a + b)² = (a + b)(a + b). Use FOIL to expand and confirm you get a² + 2ab + b².