Instantly expand (a+b)² or factor perfect square trinomials. Use our Square of a Binomial Calculator for step-by-step math solutions and geometric proofs.
Square of a Binomial Calculator: Expand & Factor Instantly Algebra can seem tricky. Seeing (3x + 4)², you might want to just square the numbers. But you know there is a specific rule to follow.…
Algebra can seem tricky. Seeing (3x + 4)², you might want to just square the numbers. But you know there is a specific rule to follow. Or maybe you are staring at x² + 6x + 9 and trying to turn it back into a bracket.
Are you a student needing to pass a test? Maybe a parent helping with homework? You are in the right place. Our Square of a Binomial Calculator is the best tool for the job.
This tool at My Online Calculators is unique. It does more than give answers. It solves two problems: it expands binomial squares and it factors perfect square trinomials. Plus, our visual tool helps you see why the math works.
First, let’s understand the math. In algebra, a binomial has two terms. These terms are separated by a plus or minus sign. Examples are (a + b) or (2x – 5).
To square a binomial, you multiply it by itself. It looks like this:
The result is a perfect square trinomial. A trinomial has three terms. Seeing the link between the question (binomial) and the answer (trinomial) is a key skill. It helps you solve harder math problems later on.
We built this tool to be easy and accurate. It has two modes to help you solve different problems. Here is how to use them.
Use this when you have a short binomial, like (x + 5)², and need the full equation.
Use this when you have a long equation and want to condense it.
You should memorize the formulas for exams. There are two main versions.
When adding terms:
(a + b)² = a² + 2ab + b²
When subtracting terms:
(a – b)² = a² – 2ab + b²
The middle term is negative (-2ab). The last term (b²) is always positive.
Students often ask, “Why isn’t the answer just a² + b²?” The best answer uses shapes.
Imagine a big square with sides of length (a + b). The total area is (a + b)². If you slice this square, you get four pieces:
To get the total area, you add them all up: a² + b² + ab + ab. This simplifies to a² + 2ab + b². If you forget the 2ab, you are missing two whole rectangles!
Let’s try three examples by hand.
Problem: Expand (x + 4)²
Problem: Expand (3y – 5)²
Problem: Expand (2x² + 3y³)²
You may know the FOIL method (First, Outer, Inner, Last). It is used for multiplying any two binomials. You can use a multiplying polynomials calculator to check this work, but knowing the shortcut is faster.
The squaring formula is just a shortcut for FOIL. Since the “Outer” and “Inner” steps are the same, we just do one calculation and double it. FOIL is a general tool; the squaring formula is a specialized tool.
Factoring is the reverse of expanding. You start with the answer and find the question. This is useful for solving equations, similar to what you might do with a factoring trinomials calculator.
Is your equation a perfect square? Check these three things:
If it passes, check the middle. Take the roots of the first and last numbers. Multiply them and double the result. Does it match the middle term?
Example: x² + 10x + 25
Watch out for these common errors:
Math shows up in real life more than you think.
A binomial has two terms. A trinomial has three. Squaring a binomial creates a trinomial.
No. This tool is for two terms only. Three terms require a longer formula.
Yes. Squaring removes the negative sign, so the answer is the same.
It comes from adding the two identical rectangles in the area model. In FOIL, it is the sum of the Outer and Inner steps.
Mastering the square of a binomial helps with algebra, calculus, and real-world science. Whether you use the formula, the visual model, or our calculator, understanding the pattern is key. Bookmark this page for the next time you need to expand or factor quickly!
It expands a binomial multiplied by itself, like (a + b)² or (a - b)², and gives you the equivalent trinomial.
Most calculators use one of these identities:
(a + b)² = a² + 2ab + b²(a - b)² = a² - 2ab + b²So instead of leaving the expression in factored form, it returns the expanded form.
Because the multiplication creates three “parts” every time:
a²2ab (this is the middle term)b²Even when the terms look simple, the middle term doesn’t disappear unless one term is 0.
Use parentheses. They tell the calculator what you want squared.
Good inputs:
(x + 3)^2(2y - 5)^2Common input that causes mistakes:
x + 3^2 (this usually means x + 9, not (x + 3)²)If the calculator supports it, the ² symbol is fine too, as long as the binomial is in parentheses.
(a - b)²?Forgetting that only the middle term changes sign.
This is the correct pattern:
(a - b)² = a² - 2ab + b²A lot of people accidentally square each term and stop there, getting a² - b², but that’s a different identity and it’s not the same expression.
Quick check with an example:
(x - 3)² = x² - 6x + 9, not x² - 9No, not when you plug in real numbers and evaluate it. A square is always zero or positive.
For example, if you evaluate (a + b)² at specific values of a and b, the final value can’t be negative. The expanded form might include a negative middle term (like -2ab), but the whole expression still represents a square.
It can do either, and you’ll usually get the same result.
a² ± 2ab + b²(a ± b)(a ± b) term by termIf you’re checking work, it helps to know both. The shortcut is faster, FOIL is easier to trust when you’re first learning.
Sometimes, yes, but only when the trinomial is a perfect square trinomial, meaning it matches one of these forms:
a² + 2ab + b² factors to (a + b)²a² - 2ab + b² factors to (a - b)²A quick clue is the first and last terms being perfect squares, and the middle term matching 2ab (with the correct sign).
Sure. If you enter (2y + 5)², the expanded result should be:
4y² + 20y + 25And if you enter (2y - 5)², the expanded result should be:
4y² - 20y + 25That sign change in the middle term is the main thing to watch.