Lesson The square of the sum formula

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The square of the sum formula


This lesson is focused on the very useful formula of the square of a sum of two numbers.
This formula is

%28a+%2B+b%29%5E2+=+a%5E2+%2B+2ab+%2B+b%5E2.

The formula is valid for any real numbers a and b.
In particular, it is valid for all integer numbers.
The formula is valid for the complex numbers as well.

The proof of the formula is very simple. It follows straightforward from the direct calculations:

.

As you see, the distributive and the commutative properties of addition and multiplication operations over the real numbers are used in derivation the formula.

The square of the sum formula is useful in a number of applications.
In many cases, you can apply it to perform simple calculations mentally without using paper and pencil (or calculator).

Example 1

Apply the square of the sum formula to calculate 11%5E2.

Solution

You have
11%5E2+=+%2810%2B1%29%5E2+=+10%5E2+%2B+2%2A10%2A1+%2B+1%5E2+=+100+%2B20+%2B1+=+121.

Example 2

Apply the square of the sum formula to calculate 21%5E2.

Solution

You have
21%5E2+=+%2820%2B1%29%5E2+=+20%5E2+%2B+2%2A20%2A1+%2B+1%5E2+=+400+%2B40+%2B1+=+441.

Example 3

Calculate 31%5E2.

Solution

Apply the square of the sum formula. You have
31%5E2+=+%2830%2B1%29%5E2+=+30%5E2+%2B+2%2A30%2A1+%2B+1%5E2+=+900+%2B60+%2B1+=+961.

Do yourself

Apply the square of the sum formula to check that

41%5E2+=+1681,
51%5E2+=+2601,
101%5E2+=+10201.

The square of the sum formula has a remarkable geometric illustration.
It is presented in the Figure below.


Figure. Illustration to the square of the sum formula

The Figure shows the big square with the side length a%2Bb subdivided into four parts by red straight lines. Two of the parts are squares with the side sizes a and b.
Two other parts are congruent rectangles with the side sizes a and b each. The areas of two smaller squares are a%5E2 and b%5E2 respectively, while the area
of the large square is %28a%2Bb%29%5E2. The area of each of the two rectangles is a%2Ab. Since the area of the large square is the sum of the areas of its parts, the Figure illustrates
the square of the sum formula

%28a+%2B+b%29%5E2+=+a%5E2+%2B+2ab+%2B+b%5E2.

You should memorize this formula. The illustration could help you memorize it.


The square of the sum formula is applicable not only to numbers. It is applicable for binomials too. For example,

%28cx%5E3+%2B+dx%29%5E2+=+c%5E2%2Ax%5E6+%2B+2cd%2Ax%5E4+%2B+d%5E2%2Ax%5E2.

Here cx%5E3 and dx are monomials that you can treat like the symbols a and b in the square of the sum formula.
You can check validity of the last formula directly by performing all relevant calculations: opening the brackets, multiplying the terms and combining the like terms.
You will get the same result. It is not surprising, because addition and multiplication operations over polynomials have the same distributive and commutative properties
as over the real numbers. Thus, the square of the sum formula is simply the useful shortcut formula.
It may help you in different ways when you need to simplify the polynomial expressions or to factor polynomials.

Example 4

Simplify the expression x%5E4%2B4x%5E2%2B4.

Solution

Apply the square of the sum formula. You have

x%5E4+%2B+4x%5E2+%2B+4+=+%28x%5E2%29%5E2+%2B+2%2Ax%5E2+%2B+2%5E2+=+%28x%5E2+%2B+2%29%5E2.


Do yourself

Apply the square of the sum formula to factor

4x%5E4+%2B+4x%5E2+%2B+1,

a%5E4+%2B+2a%5E2%2Ab%5E2+%2B+b%5E4.


Example 5

Simplify the expression 49a%5E6%2Ab%5E2+%2B+42a%5E3%2Ab+%2B+9.

Solution

Apply the square of the sum formula. You get

.

Note. Pay attention how the terms are grouped to follow the pattern of the square of the sum formula.


Example 6

Factor the trinomial 9a%5E4+%2B+30a%5E2%2Ab%5E3+%2B+25b%5E6.

Solution

Apply the square of the sum formula. You get

.

Note. Pay attention how the brackets are used to group monomials according the pattern of the square of the sum formula.


Example 7

Factor the trinomial 27a%5E5%2Ab+%2B+90a%5E3%2Ab%5E4+%2B+75ab%5E7.

Solution

First take the common factor 3ab out the brackets. You get

.

Now, factor the trinomial 9a%5E4+%2B+30a%5E2%2Ab%5E3+%2B+25b%5E6, which is in the brackets on the right side, as it is done in the Example 6 above.
Finally, you get the required factorization

27a%5E5%2Ab+%2B+90a%5E3%2Ab%5E4+%2B+75ab%5E7+=+3ab%2A%283a%5E2+%2B+5b%5E3%29%5E2.


Summary

The square of the sum formula

%28a+%2B+b%29%5E2+=+a%5E2+%2B+2ab+%2B+b%5E2

is useful shortcut multiplication formula.
At the same time, you can use it to factor trinomials when applicable.


For similar lessons see
    The square of the difference formula and
    The difference of squares formula
under the current topic in this site.

For the list of all shortcut quadratic multiplication formulas see the lesson
    OVERVIEW of shortcut quadratic multiplication formulas
under the current topic in this site.

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