Square of a Binomial Calculator

The square of a binomial calculator is here for you whenever you have trouble multiplying a binomial by itself. Everybody knows and studies them in high school, but they get complicated as numbers become bigger. That’s why we’ve created this tool to simplify the process for you!

In this article, we will discuss the following topics:

• What is the square of a binomial?
• What is the binomial squared formula?
• How to use our square of a binomial calculator.

If you don't feel like reading the whole article, skip to the FAQ section for a quick recap. There, you’ll find answers to questions like How do you square a binomial difference? and What is the result when you square a binomial?

The result of the square of a binomial is called a perfect square trinomial. The rule for the square of a binomial is pretty easy. Just follow these steps:

1. Take your binomial in the form (a + b)².

2. Take the first term of your binomial and raise it to the power of 2. That's the first term of your result: a².

3. For the second term of your result, multiply a by b and double the result: 2ab.

4. Lastly, raise b to the power of 2. That's your third and final term.

5. Combine the obtained terms with + signs and/or − signs between them.

And that's the job done! The binomial squared formula is as follows:

(a + b)² = a² + 2ab + b²

The square of a binomial calculator has two functionalities, so it is divided into two sections. Let's take a look at the first one.

Simple binomial expansion

The first section of the calculator allows you to expand your expression given one of the terms. Let's take an example. Say you have to expand the following expression: (6 − b)².

1. Choose the form of your binomial. We will select the second option, (a − b)².

   Either way, you can input both positive and negative numbers, so the first term can also be negative if needed.

2. Choose the known term. In this case, we will leave it as option a.

3. Insert the known term: 6.

4. The calculator will not only provide you with your result, but it will also give you the steps necessary to obtain it. Here is our step-by-step binomial expansion:
   
   (6 − b)² =

   6² − (2 × 6 × b) + b² =

   36 − 12b + b²

5. Optional: Select Solve for the unknown variable and enter the result of your expression. Let's input 25.

6. According to the square of a binomial calculator, our missing variable must be equal to 1 or 11 in order for the equation to be true.

Expansion of (ax + b)² given a and b

Let's now see the steps for the second section of our tool, which you can expand by selecting the second option at the top of the tool.

1. Insert your a and b. In this example, we will use the following expression: (17x + 210)². Those numbers will now get a bit more complicated!

2. As soon as you enter your data, you will see the following information:

   • The step-by-step binomial expansion

     y = (17x + 210)²

     y = (17x)² + (2 × 17 × 210)x + 210²

     y = 289x² + 7,140x + 44,100

   • The x-intercept of the obtained function (unless your a is equal to 0)

     In this case, the x-intercept of our function is (-12.353, 0).

   • The y-intercept of the obtained function

     The y-intercept is (0, 44,100).

3. Optional (if your a is not equal to 0): Click Solve for a given y and insert a value for y. In this case, let's input 345.

4. The calculator will display the two roots of your equation.

   Here, we have:

   289² + 7,140x + 44,100 = 345

   289² + 7,140x + 43,755 = 0

   Therefore,

   • x₁ = -11.26; and
   • x₂ = -13.446.

 
Note that these results are approximated. If you need your result to be more accurate, use the following formulae:

• For every quadratic formula in the form ax² + bx + c = 0,

  x₁ = [-b + √(b² − 4ac)] / 2a; and

  x₂ = [-b − √(b² − 4ac)] / 2a.