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.
Try the free calculator
Put these formulas into practice with our instant, step-by-step Square of a Binomial Calculator.
Use our free **Square of Binomial Calculator** to **expand binomial squares** instantly. This **binomial square calculator** helps you compute both **(a + b)²** and **(a - b)²** with step-by-step expansion and verification. Whether you need a **perfect square formula** calculator for homework or an **algebra binomial square tool** for quick reference, this tool provides accurate results with detailed FOIL breakdowns. 100% free — no signup required!
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
- Step 1: Enter the value or expression for a (first term of the binomial) into the **square of binomial calculator**.
- Step 2: Enter the value or expression for b (second term of the binomial).
- Step 3: Review the expanded forms for both **(a + b)²** and **(a - b)²** displayed side by side.
- Step 4: Check the **step-by-step FOIL expansion** showing how each term is computed.
- Step 5: Verify the **perfect square trinomial** result by confirming the middle term is ±2ab.
- Step 6: Use the verification step to confirm the expansion by multiplying the binomial by itself.
- Step 7: Apply the results to simplify algebraic expressions or solve quadratic equations.
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. Our **binomial square calculator** shows the full step-by-step expansion.
Is this square of binomial calculator free to use?
Yes, this **free binomial square calculator online** is 100% free with no signup required. It provides instant results with step-by-step FOIL breakdowns for both (a + b)² and (a - b)².
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. Our **binomial expansion calculator** shows this derivation clearly.
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. The **perfect square trinomial calculator** helps avoid this mistake.
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. Our **algebra binomial square tool** handles expressions with variables automatically.
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. The **binomial squared step by step** breakdown shows each FOIL step.
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. Our **square of sum and difference calculator** computes both forms side by side.
What is a perfect square trinomial?
A perfect square trinomial is the result of squaring a binomial: a² + 2ab + b² or a² - 2ab + b². It can be factored back into (a + b)² or (a - b)². Recognizing perfect square trinomials is key to factoring quadratic expressions.
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. For higher powers, use the binomial theorem.
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. Our **quadratic expansion calculator** includes a built-in verification step.