What Is Binomial Squared?
A binomial squared is a binomial multiplied by itself. The result follows a fixed pattern: (a + b)² = a² + 2ab + b². There are always three terms, and the middle one, 2ab, is the part people forget.
Type a binomial such as 3x − 2, choose squared or cubed, and the calculator shows A², 2AB and B² separately, then the expanded result. You can also evaluate it at a value to confirm the original and the expansion agree.
If the first and last terms of a trinomial are perfect squares and the middle term is twice their product, the trinomial is a perfect square and factors back into a squared binomial.
How to Use the Binomial Squared Calculator
- Type the binomial, for example 3x − 2.
- Choose squared or cubed.
- Read the three parts and the expanded result.
Binomial Squared Formula
Binomial squared
Difference squared
Binomial cubed
Binomial Squared Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| A | The first term of the binomial | |
| B | The second term of the binomial | Keep its sign |
| A² | First term squared | |
| 2AB | Twice the product of the two terms | The middle term |
| B² | Second term squared | Always positive |
Binomial Squared Worked Example
With the example values loaded in the calculator above, the steps are:
- Write the binomial as (A + B) with A = 3x and B = −2.
- A² = 9x².
- 2AB = 2 × (3x) × (−2) = −12x.
- B² = 4.
- (A + B)² = A² + 2AB + B² = 9x² − 12x + 4.
Result: (3x − 2)² expanded = 9x² − 12x + 4.
Worked example: (3x − 2)²
A = 3x and B = −2. A² = 9x². 2AB = 2 × 3x × (−2) = −12x. B² = 4.
The result is 9x² − 12x + 4. At x = 2, (3 × 2 − 2)² = 16 and 36 − 24 + 4 = 16.
Where Binomial Squared Is Used
Expanding in algebra
Check (3x − 2)² or (2y + 5)² without errors.
Recognising perfect squares
See if a trinomial matches A² ± 2AB + B².
Mental maths
Use the pattern to square numbers such as 51² = (50 + 1)².
Common Binomial Squared Mistakes to Avoid
- The sign of the middle term follows the sign between the terms.
- If the first and last terms are perfect squares and the middle is twice their product, the trinomial is a perfect square.
Binomial Squared Terms and Definitions
- Perfect square trinomial
- A trinomial that factors as the square of a binomial, such as 9x² − 12x + 4.
Binomial Squared FAQs
What is the formula for a binomial squared?
(a + b)² = a² + 2ab + b².
Why is there a middle term?
Multiplying (a + b)(a + b) produces ab twice, once from each cross product.
How do I square a binomial with a negative term?
Use (a − b)² = a² − 2ab + b². The last term is always positive.
How many terms does the answer have?
Three, unless some combine or cancel.
Can I cube a binomial here?
Yes. Choose power 3 to use (a + b)³ = a³ + 3a²b + 3ab² + b³.
How do I check the answer?
Substitute a number into the original binomial, square it, and compare with the expanded answer at that number.