What Is Square of a Binomial?
The square of a binomial is the product of a binomial with itself. It follows the pattern (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². The result is a perfect square trinomial with three terms, not two.
Enter the coefficient and the second term, choose plus or minus, and the calculator expands the square, shows each part (first term squared, twice the product, last term squared), compares it with the common mistake a² + b², and draws an area model. The second mode takes a trinomial Ax² + Bx + C and checks whether B² = 4AC so it factors as a squared binomial.
Below are the formulas, a variables table, a worked example and answers to common questions.
How to Use the Square of a Binomial Calculator
- Choose to expand a binomial or test a trinomial.
- Enter the coefficient and the second term, and choose + or −.
- Read the expanded trinomial and the steps.
- Use the area model to see where each term comes from.
Square of a Binomial Formula
Square of a sum
Square of a difference
Perfect square test
Square of a Binomial Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a | The first term of the binomial | |
| b | The second term of the binomial | |
| 2ab | The middle term | twice the product of the two terms |
| B² = 4AC | Test for a perfect square trinomial |
Square of a Binomial Worked Example
Expand (3x + 5)². The first term squared is (3x)² = 9x². The middle term is 2 × 3x × 5 = 30x. The last term squared is 5² = 25. So (3x + 5)² = 9x² + 30x + 25. Check at x = 1: (3 + 5)² = 64 and 9 + 30 + 25 = 64.
With the example values loaded in the calculator above, the steps are:
- Use (A + B)² = A² + 2AB + B², with A = 3x and B = 5.
- A² = (3x)² = 9x².
- 2AB = 2 × 3x × 5 = 30x.
- B² = 5² = 25.
- Add them: 9x² + 30x + 25.
Result: (3x + 5)² expanded = 9x² + 30x + 25; Perfect square trinomial? = Yes: the result is a perfect square trinomial.
How to Read the Expansion
(a + b)² = a² + 2ab + b². For (3x + 5)² the first term is 9x², the middle term is 2 × 3x × 5 = 30x and the last term is 25. So (3x + 5)² = 9x² + 30x + 25. At x = 1 the check is 8² = 64 and 9 + 30 + 25 = 64.
The common mistake is (a + b)² = a² + b², which drops the middle term 2ab. The area model shows why: a square with side a + b contains two a × b rectangles in addition to the two squares.
Where Square of a Binomial Is Used
Expanding
Square a binomial with a coefficient and a sign.
Perfect square trinomials
Check whether Ax² + Bx + C is a perfect square.
Algebra practice
See the first, middle and last terms separately.
Checking answers
Use the evaluation at a chosen x.
Common Square of a Binomial Mistakes to Avoid
- Do not forget the middle term 2ab.
- For (a − b)² the middle term is negative: a² − 2ab + b².
- Square the coefficient as well as the variable: (3x)² = 9x².
- A trinomial Ax² + Bx + C is a perfect square when B² = 4AC and A and C are perfect squares.
Square of a Binomial Terms and Definitions
- Binomial
- An expression with two terms.
- Perfect square trinomial
- A trinomial that is the square of a binomial.
Square of a Binomial FAQs
What is the square of a binomial?
It is the result of multiplying a binomial by itself. (a + b)² = a² + 2ab + b², and (a − b)² = a² − 2ab + b².
Is (a + b)² equal to a² + b²?
No. The expansion has a middle term, 2ab. For a = 3 and b = 5, (a + b)² is 64 and a² + b² is only 34.
How do I square a binomial with a coefficient?
Square the whole first term and the whole last term, and double their product. For (3x + 5)², the terms are 9x², 30x and 25.
How can I tell if a trinomial is a perfect square?
Check that the first and last terms are perfect squares and that the middle term is twice the product of their roots, with the right sign.