Explainer: Squaring Binomials

In this explainer, we will learn how to expand the square of the difference or sum of two monomials.

Recall that a binomial is an expression that contains exactly two monomials (or terms). If we want to find the square of a binomial, we multiply it by itself. For example, consider the binomial 𝑥+2; the square of 𝑥+2 would then be written as (𝑥+2) which is exactly the same as (𝑥+2)(𝑥+2).

At this point, we can expand this expression using a variety of different methods. Firstly, we can use the distributive property to rewrite the expression as (𝑥)(𝑥+2)+(2)(𝑥+2).

If we then expand each of the two parentheses, again using the distributive property, we get 𝑥+2𝑥+2𝑥+4.

Finally, collecting like terms, we get 𝑥+4𝑥+4.

Two alternative methods for expanding this expression are to use arrows to indicate all the multiplications or use a grid to keep track of them. The first method is shown below.

This method is sometimes referred to as the FOIL method. This is because we consider the four terms we get from the products of the

  • First terms,
  • Outer terms,
  • Inner terms,
  • Last terms.

The multiplication grid method is shown below.

The multiplication grid can be visualized using an area model which gives a clearer visual intuition as to where the different terms come from in the formula. We can think of (𝑥+2) as representing the area of a square of side length 𝑥+2. We can then break the area down into the parts contributed by the combinations of 𝑥 and 2 as represented below.

As we can see, the total area (𝑥+2) is made up of the sum of the areas of the four sections which are 𝑥, 2𝑥, 2𝑥, and 2. Hence, (𝑥+2)=𝑥+4𝑥+4.

Let us now look at the square of a general binomial. If we consider the expression (𝑎+𝑏), this is the same as (𝑎+𝑏)(𝑎+𝑏).

If we then expand the brackets, we get 𝑎+𝑎𝑏+𝑏𝑎+𝑏.

Given that multiplication is commutative, we have two like terms, so the expression simplifies to 𝑎+2𝑎𝑏+𝑏.

This gives us the general form of the square of a binomial. For example, consider the expression (𝑥+5); here, we have that 𝑎=𝑥 and 𝑏=5. The expansion will, therefore, be equal to (𝑥)+2(5)(𝑥)+(5), which simplifies to 𝑥+10𝑥+25.

We do need to be a little careful if either 𝑎 or 𝑏 is negative as this will affect the sign of the middle term.

Let us now look at a series of examples.

Example 1: Squaring Binomials Where Both Terms Are Positive

Expand (𝑚+4).

Answer

Our first step is to remember that (𝑚+4)=(𝑚+4)(𝑚+4).

If we then expand the expression, we get 𝑚+4𝑚+4𝑚+16, which simplifies to 𝑚+8𝑚+16.

Example 2: Squaring Binomials Where One Term Is Negative

Expand (𝑚4).

Answer

Our first step is to remember that (𝑚4)=(𝑚4)(𝑚4).

If we then expand the expression, being particularly careful with signs, we get 𝑚4𝑚4𝑚+16, which simplifies to 𝑚8𝑚+16.

Example 3: Squaring Binomials with More Complicated Terms

Expand (2𝑥+5).

Answer

Our first step is to remember that (2𝑥+5)=(2𝑥+5)(2𝑥+5).

If we then expand the expression, we get 4𝑥+10𝑥+10𝑥+25, which simplifies to 4𝑥+20𝑥+25.

Example 4: Squaring Binomials with More Complicated Terms

Expand 4𝑥+32𝑦.

Answer

Our first step is to remember that 4𝑥+32𝑦=4𝑥+32𝑦4𝑥+32𝑦.

If we then expand the expression, we get 16𝑥+6𝑥𝑦+6𝑥𝑦+94𝑦, which simplifies to 16𝑥+12𝑥𝑦+94𝑦.

Example 5: Squaring Binomials with More Complicated Terms

If (𝑥+𝑦)=100 and 𝑥𝑦=20, what is the value of 𝑥+𝑦?

Answer

Since we have been given the value of the square of the binomial 𝑥+𝑦, a good place to start would be to expand the binomial. Hence, (𝑥+𝑦)=𝑥+𝑥𝑦+𝑥𝑦+𝑦=𝑥+2𝑥𝑦+𝑦.

We can now substitute in the values (𝑥+𝑦)=100 and 𝑥𝑦=20 to get 100=𝑥+2(20)+𝑦.

We can then rearrange this formula to make 𝑥+𝑦 the subject by subtracting 40 from both sides of the equation as follows: 𝑥+𝑦=60.

Example 6: Squaring Binomials with More Complicated Terms

Given that 𝑥+𝑦=77 and 𝑥𝑦=14, what are the possible values of 𝑥𝑦?

Answer

To answer this question, we need to connect 𝑥𝑦 to 𝑥+𝑦 and 𝑥𝑦. Since we have terms, square terms, and cross terms, a good place to start is by considering the square of the binomial 𝑥𝑦. Therefore, expanding (𝑥𝑦), we have (𝑥𝑦)=𝑥𝑥𝑦𝑥𝑦+𝑦=𝑥2𝑥𝑦+𝑦=𝑥+𝑦2𝑥𝑦.

We can now substitute 𝑥+𝑦=77 and 𝑥𝑦=14 into this equation to get (𝑥𝑦)=772(14)=49.

By taking square roots and considering both the positive and the negative values, we have that 𝑥𝑦=7 or 𝑥𝑦=7.

Key Points

  1. We can expand the squares of binomials using different techniques such as the following:
    1. The FOIL method, where we consider the four terms we get from the products of the
      • First terms,
      • Outer terms,
      • Inner terms,
      • Last terms.
      We can often use arrows to keep track of this as follows.
    2. The grid method, where we write the two terms of the binomial as the rows and columns on a table and then take the product of each pair, as shown in the example below.
  2. The general formula for expanding the square of a binomial is (𝑎+𝑏)=𝑎+2𝑎𝑏+𝑏. If we are using this formula, we need to be careful with our minus signs if either 𝑎 or 𝑏 is negative.

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