Square Binomial Calculator

Master binomial squares with instant expansion, factoring, and term breakdown and verification. Adjust coefficients, select plus or minus, and preview dynamic algebra in action. See A², 2AB, B² components, with clean steps and explanations included. Export calculations and tables as CSV or PDF files.

Inputs
A = m·x, B = n·y

Form covers (m·x ± n)² or (m·x ± n·y)² with symbolic and numeric outputs.

More Options
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Results
No results yet. Enter values and press Compute.
Expression Expanded form Term A² Term 2AB Term B² Coeff x² Coeff x Coeff xy Constant Evaluated value
Steps
    Batch Mode

    Paste lines as: m,sign,n or m,sign,n,twoVar,y. Example: 3,-,4 or 2,+,0.5,1,1.

    Example Data
    # m sign n twoVar y Expression Expanded
    1 1 + 5 0 - (x + 5)² x² + 10x + 25
    2 3 − 4 0 - (3x − 4)² 9x² − 24x + 16
    3 2 + 1 1 y (2x + 1·y)² 4x² + 4xy + y²
    Formula Used

    (A ± B)² = A² ± 2AB + B²

    Let A = m·x and B = n or n·y. Then:

    • A² = (m·x)² = m²·x²
    • 2AB = 2·(m·x)·(n or n·y) = 2mn·x or 2mn·x·y
    • B² = (n)² or (n·y)² = n² or n²·y²

    Minus case: (m·x − B)² = m²x² − 2mBx + B², matching input choice.

    How to Use
    1. Set symbols, coefficients, and choose plus or minus.
    2. Toggle second variable to expand (m·x ± n·y)² as needed.
    3. Optionally enter values for x and y to evaluate numerically.
    4. Enable steps for a transparent derivation of the expansion.
    5. Export the results table as CSV or print to PDF when ready.

    Tip: Share settings using the live link or save inputs locally.

    Additional Insights
    1) Properties and Identities
    • (a + b)² = a² + 2ab + b²
    • (a − b)² = a² − 2ab + b²
    • Always non‑negative for real a, b: squares are ≥ 0.
    • Perfect square trinomials factor back to (a ± b)².
    2) Common Patterns and Special Cases
    Input form Expanded result Notes
    (x + 1)² x² + 2x + 1 Unit increment; linear coefficient equals 2
    (kx + k)² k²x² + 2k²x + k² Factor out k²: k²(x + 1)²
    (mx − n)² m²x² − 2mnx + n² Middle term sign flips to negative
    (2x + y)² 4x² + 4xy + y² Two‑variable case; xy mixed term appears
    3) Worked Examples
    # Expression Steps (A², 2AB, B²) Expanded
    1 (3x + 2)² A² = 9x², 2AB = 12x, B² = 4 9x² + 12x + 4
    2 (x − 7)² A² = x², 2AB = −14x, B² = 49 x² − 14x + 49
    3 (2x + 5y)² A² = 4x², 2AB = 20xy, B² = 25y² 4x² + 20xy + 25y²
    4) Practical Uses and Tips
    • Detect perfect square trinomials in factoring problems quickly.
    • Complete the square when solving quadratics or optimization tasks.
    • Estimate values: (x + δ)² ≈ x² + 2xδ + δ² for small δ.
    • In geometry, models area of a square with side (a ± b).
    FAQs
    What does this calculator compute?

    It expands and optionally evaluates squares of binomials like (m·x ± n)² or (m·x ± n·y)², showing A², 2AB, B² terms, coefficients, and a clean, readable expanded expression.

    How do plus and minus affect the middle term?

    For (A + B)², the middle term is +2AB. For (A − B)², the middle term is −2AB. Squares A² and B² are always positive contributions in the expanded expression.

    Can it handle two variables?

    Yes. Enable the second-variable option to expand (m·x ± n·y)². The result includes an xy mixed term with coefficient ±2mn, plus x² and y² terms with coefficients m² and n² respectively.

    What’s the difference between expansion and evaluation?

    Expansion gives the symbolic polynomial form with coefficients. Evaluation substitutes numeric values for variables (x, and optionally y) to compute a single numeric result for the squared binomial.

    What mistakes should I avoid?

    Don’t forget the middle term 2AB. Watch sign selection for plus versus minus cases. If evaluating numerically, ensure decimal places and input values are set correctly for the desired precision.