Square of Sum || ( a + b ) square formula

I this chapter we will learn square of sum formula and will solve some examples related to the formula.

Square of Sum formula

If a & b are two entities then square of a + b is given by following formula.

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

Hence of square of sum (a + b) is equal to sum of square of individual terms & 2ab.

The formula is very important as it would help us to solve different algebra problems. I urge you to memorize this formula for future use.

Derive Square of Sum equation

Let a & b are the two terms.

The square of sum of a & b can be written as;

\mathtt{( a\ +\ b)^{2} =\ ( a+b) .( a+b)}

Simplifying the equation further.

\mathtt{( a\ +\ b)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( a+b) .( a+b)}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +ab+ba+b^{2}}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +\ 2ab\ +\ b^{2}}

Hence, the formula derived is;

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

Note:
The formula is also called square of binomial formula as the term (a + b) is a binomial with two terms.

Proof of square of sum formula

Consider the number \mathtt{( 9+2)^{2}}

Finding the value of number using simple calculation.

\mathtt{\Longrightarrow \ ( 9+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 11^{2}}\\\ \\ \mathtt{\Longrightarrow \ 11.\ 11}\\\ \\ \mathtt{\Longrightarrow \ 121}

Hence, 121 is the required value.

Now find the value of number using square of sum formula.

Using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{\Longrightarrow \ ( 9+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 9^{2} +2( 9)( 2) \ +2^{2}}\\\ \\ \mathtt{\Longrightarrow \ 81+\ 36+4}\\\ \\ \mathtt{\Longrightarrow \ 121}

Using the formula we get the same value 121.

Hence, the above formula is valid.

Geometric Representation of Sum of square formula

Consider the square of length (a + b)
Then area of square is \mathtt{( a\ +\ b)^{2}}

The above square contain 4 parts.

Part (i) ⟹ Square with area \mathtt{a^{2}}

Part (ii) ⟹ Rectangle with area a.b

Part (iii) ⟹ Square with area \mathtt{b^{2}}

Part (iv) ⟹ Rectangle with area a.b

Combining all the parts we get;

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

Square of Sum – Solved Problems

Example 01
Expand \mathtt{( 9x\ +\ 2y)^{2}}

Solution
The expression is in form of square of sum.

We will use formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{( 9x+2y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 9x)^{2} +2.9x.2y+( 2y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 81x^{2} +36xy\ +4y^{2}}

Hence, \mathtt{81x^{2} +36xy\ +4y^{2}} is the expanded form of expression.

Example 02
Expand \mathtt{( 7x\ +\ 10)^{2}}

Solution
Using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{( 7x\ +\ 10)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 7x)^{2} +2.7x.10+( 10)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 49x^{2} +140x\ +100}

Hence, \mathtt{49x^{2} +140x\ +100} is the expanded form of expression.

Example 03
Find value of \mathtt{( 8.1)^{2}} using square of sum formula.

Solution
\mathtt{( 8.1)^{2}} can be written as;

\mathtt{( 8.1)^{2} \ \Longrightarrow \ ( 8+0.1)^{2}}

Now using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{( 8\ +\ 0.1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 8)^{2} +2( 8)( 0.1) +( 0.1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 64+1.6\ +0.01}\\\ \\ \mathtt{\Longrightarrow \ 65.61}

Hence, 65.61 is the value of given number.

Example 04
Find the value of \mathtt{( 12.5)^{2}} using square of sum formula.

Solution
The number can be written as;
\mathtt{( 12.5)^{2} \ \Longrightarrow \ ( 12+0.5)^{2}}

Now using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{( 12\ +\ 0.5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 12)^{2} +2( 12)( 0.5) +( 0.5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 144+12\ +0.25}\\\ \\ \mathtt{\Longrightarrow \ 156.25}

Hence, 156.25 is the value of given expression.

Example 05
Find the value of \mathtt{(14)^{2}} using square of sum formula.

Solution
The number can be written as;
\mathtt{( 14)^{2} \ \Longrightarrow \ ( 10+4)^{2}}

Using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values we get;
\mathtt{( 10\ +\ 4)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 10)^{2} +2( 10)( 4) +( 4)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 100+80\ +16}\\\ \\ \mathtt{\Longrightarrow \ 196}

Hence, 196 is the value of given expression.

Example 06
Expand \mathtt{\left( 3x^{2} \ +\ 2\right)^{2}}

Solution
Using the formula;
\mathtt{( a\ +\ b)^{2} =\ a^{2} +\ 2ab\ +\ b^{2}}

Putting the values;
\mathtt{\left( 3x^{2} \ +\ 2\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( 3x^{2}\right)^{2} +2\left( 3x^{2}\right)( 2) +( 2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 9x^{4} +12x^{2} \ +\ 4}

Hence, \mathtt{9x^{4} +12x^{2} \ +\ 4} is the expanded form of expression.