Lesson The square of the sum formula

Source code of 'The square of the sum formula'

This Lesson (The square of the sum formula) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

<H2>The square of the sum formula</H2>

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

This formula is

{{{(a + b)^2 = a^2 + 2ab + b^2}}}.

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:

{{{(a + b)^2 = (a + b)*(a + b) = a^2 + b*a + a*b + b^2 = a^2 + 2ab + b^2}}}.

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 <B><I>square of the sum formula</I></B> 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).

<H3>Example 1</H3>Apply the <B><I>square of the sum formula</I></B> to calculate {{{11^2}}}.

<B>Solution</B>

You have

{{{11^2 = (10+1)^2 = 10^2 + 2*10*1 + 1^2 = 100 +20 +1 = 121}}}.

<H3>Example 2</H3>Apply the <B><I>square of the sum formula</I></B> to calculate {{{21^2}}}.

<B>Solution</B>

You have

{{{21^2 = (20+1)^2 = 20^2 + 2*20*1 + 1^2 = 400 +40 +1 = 441}}}.

<H3>Example 3</H3>Calculate {{{31^2}}}.

<B>Solution</B>

Apply the <B><I>square of the sum formula</I></B>. You have

{{{31^2 = (30+1)^2 = 30^2 + 2*30*1 + 1^2 = 900 +60 +1 = 961}}}.

<H3>Do yourself</H3>Apply the <B><I>square of the sum formula</I></B> to check that

{{{41^2 = 1681}}},
{{{51^2 = 2601}}},
{{{101^2 = 10201}}}.

The <B><I>square of the sum formula</I></B> has a remarkable geometric illustration.
It is presented in the <B>Figure</B> below.

<TABLE cellspacing="20"> 
 <TR>
 <TD>
{{{drawing( 240, 240, -1, 5, -1, 5,

line   (0, 0, 5, 0),
            line   (5, 0, 5, 5),
            line   (0, 5, 5, 5),
            line   (0, 0, 0, 5),

red(line   (3.5, 0,   3.5, 5)),
        red(line   (0,   3.5, 5,   3.5)),

locate (1.7, 0.4, a),
            locate (4.1, 0.4, b),
            locate (2.2, -0.1, a+b),

locate (0.1,  2.0,  a),
            locate (0.1,  4.35, b),
            locate (-0.9, 2.9, a+b),

locate (1.6, 2.2, a^2),
            locate (4.0, 4.6, b^2),

locate (1.5, 4.35, ab),
            locate (4.0, 2.0,  ab)
)}}}
<B>Figure</B>. Illustration to the <B><I>square of the sum formula</I></B> 
  </TD>
  </TR>
</TABLE>

The <B>Figure</B> shows the big square with the side length {{{a+b}}} 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^2}}} and {{{b^2}}} respectively, while the area 
of the large square is {{{(a+b)^2}}}. The area of each of the two rectangles is {{{a*b}}}. Since the area of the large square is the sum of the areas of its parts, the <B>Figure</B> illustrates 
the square of the sum formula

{{{(a + b)^2 = a^2 + 2ab + b^2}}}.

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

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

{{{(cx^3 + dx)^2 = c^2*x^6 + 2cd*x^4 + d^2*x^2}}}.

Here {{{cx^3}}} and {{{dx}}} are monomials that you can treat like the symbols {{{a}}} and {{{b}}} in the <B><I>square of the sum formula</I></B>.
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 <B><I>square of the sum formula</I></B> 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.

<H3>Example 4</H3>Simplify the expression {{{x^4+4x^2+4}}}.

<B>Solution</B>

Apply the <B><I>square of the sum formula</I></B>. You have

{{{x^4 + 4x^2 + 4 = (x^2)^2 + 2*x^2 + 2^2 = (x^2 + 2)^2}}}.

<H3>Do yourself</H3>Apply the <B><I>square of the sum formula</I></B> to factor

{{{4x^4 + 4x^2 + 1}}},

{{{a^4 + 2a^2*b^2 + b^4}}}.

<H3>Example 5</H3>Simplify the expression {{{49a^6*b^2 + 42a^3*b + 9}}}.

<B>Solution</B>

Apply the <B><I>square of the sum formula</I></B>. You get

{{{49a^6*b^2 + 42*a^3*b + 9 = (7a^3*b)^2 + 2*3*(7a^3*b) + 3^2 = (7a^3*b + 3)^2}}}.

<B>Note</B>. Pay attention how the terms are grouped to follow the pattern of the <B><I>square of the sum formula</I></B>.

<H3>Example 6</H3>Factor the trinomial {{{9a^4 + 30a^2*b^3 + 25b^6}}}.

<B>Solution</B>

Apply the <B><I>square of the sum formula</I></B>. You get

{{{9a^4 + 30*a^2*b^3 + 25b^6 = (3a^2)^2 + 2*(3a^2)*(5b^3) + (5b^3)^2 = (3a^2 + 5b^3)^2}}}.

<B>Note</B>. Pay attention how the brackets are used to group monomials according the pattern of the <B><I>square of the sum formula</I></B>.

<H3>Example 7</H3>Factor the trinomial {{{27a^5*b + 90a^3*b^4 + 75ab^7}}}.

<B>Solution</B>

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

{{{27a^5*b + 90a^3*b^4 + 75ab^7 = 3ab*(9a^4 + 30a^2*b^3 + 25b^6)}}}.

Now, factor the trinomial {{{9a^4 + 30a^2*b^3 + 25b^6}}}, which is in the brackets on the right side, as it is done in the <B>Example 6</B> above.
Finally, you get the required factorization

{{{27a^5*b + 90a^3*b^4 + 75ab^7 = 3ab*(3a^2 + 5b^3)^2}}}.

<H3>Summary</H3>The <B><I>square of the sum formula</I></B>

{{{(a + b)^2 = a^2 + 2ab + b^2}}}

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

For similar lessons see

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/The_square_of_a_difference.lesson>The square of the difference formula</A> and

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/The_difference_of_squares_formula.lesson>The difference of squares formula</A>

under the current topic in this site.

For the <B>list of all shortcut quadratic multiplication formulas</B> see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Review-of-shortcut-quadratic-multiplication-formulas.lesson>OVERVIEW of shortcut quadratic multiplication formulas</A>

under the current topic in this site.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.