# Square of Binomial

In this chapter we will learn the concept and formula for square of binomial. In the end we will solve questions related to the concept.

Let us first understand the basics of binomial.

## What is Binomial?

Binomial is an algebraic expression with two entities.

The entities can be constants, variables or both.

Some examples of binomials are;

\mathtt{\Longrightarrow \ x\ +\ 6}\\\ \\ \mathtt{\Longrightarrow \ 3xy+x^{2} y}\\\ \\ \mathtt{\Longrightarrow \ 5xy^{3} z\ +\ 2xyz}

## Square of Binomial formula

There are two formulas for square of binomial.

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

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

Both the formulas are vey important so make sure to memorize each of them for your examination.

## Perfect square expression

If the expression is in the form of;

\mathtt{\Longrightarrow \ a^{2} +b^{2} +2ab}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +b^{2} -2ab}

then the expression is said is said to be perfect square expression since the above terms can be converted into \mathtt{( a+b)^{2} \&\ ( a-b)^{2}} respectively which is perfect square.

## Square of Binomial – Solved Problems

Example 01
Expand \mathtt{( 2x-6)^{2}}

Solution
The expression is in form of \mathtt{( a\ -\ b)^{2}}

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

Putting the values;

\mathtt{\Longrightarrow \ ( 2x-6)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 2x)^{2} +( 6)^{2} -2.( 2x)( 6)}\\\ \\ \mathtt{\Longrightarrow \ 4x^{2} +36-24x}

Hence, the above expression is the expanded form of given problem.

Example 02
Expand \mathtt{( x+2y)^{2} +( x-2y)^{2}}

Solution
The question involves square of two binomials.

We will solve question in two steps;

First expand \mathtt{( x+2y)^{2}}

\mathtt{\Longrightarrow \ ( x+2y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +( 2y)^{2} +2.( x) .( 2y)}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +4y^{2} +4xy}

Now Expand \mathtt{( x-2y)^{2}}

\mathtt{\Longrightarrow \ ( x-2y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +( 2y)^{2} -2.( x) .( 2y)}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +4y^{2} -4xy}

Now add both the parts;

\mathtt{\Longrightarrow \ ( x+2y)^{2} +( x-2y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +4y^{2} +\cancel{\mathtt{4xy}} +x^{2} +4y^{2} -\cancel{\mathtt{4xy}}}\\\ \\ \mathtt{\Longrightarrow \ 2x^{2} +8y^{2}}

Hence, the above expression is the solution.

Example 03
If \mathtt{x+\frac{1}{x} \ =\ 7} . Find the value of \mathtt{x^{4} +\frac{1}{x^{4}}}

Solution
Its given that; \mathtt{x+\frac{1}{x} \ =\ 7}

Squaring the equation on both the sides we get;

\mathtt{\left( x+\frac{1}{x}\right)^{2} \ =\ 7^{2}}\\\ \\ \mathtt{\left( x+\frac{1}{x}\right)^{2} \ =\ 49}

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

\mathtt{x^{2} +\frac{1}{x^{2}} +2.x.\frac{1}{x} =\ 49}\\\ \\ \mathtt{x^{2} +\frac{1}{x^{2}} \ +\ 2\ =\ 49}\\\ \\ \mathtt{x^{2} +\frac{1}{x^{2}} \ =\ 47}

Here we got equation \mathtt{x^{2} +\frac{1}{x^{2}} \ =\ 47} .

Again square the equation on both sides.

\mathtt{\left( x^{2} +\frac{1}{x^{2}}\right)^{2} \ =\ ( 47)^{2}}\\\ \\ \mathtt{\left( x^{2}\right)^{2} +\left(\frac{1}{x^{2}}\right)^{2} +2x^{2}\frac{1}{x^{2}} =2209}\\\ \\ \mathtt{x^{4} +\frac{1}{x^{4}} +2\ =2209}\\\ \\ \mathtt{x^{4} +\frac{1}{x^{4}} \ =\ 2207}

Hence, the value of \mathtt{x^{4} +\frac{1}{x^{4}}} is 2207.

Example 04
If x + y = 9 and x.y = 12. Find the value of \mathtt{x^{2} +y^{2}}

Solution
It’s given that, x + y = 9

Squaring both sides of equation, we get;

\mathtt{( x\ +\ y\ )^{2} =\ 9^{2}}\\\ \\ \mathtt{x^{2} +y^{2} +2xy\ =\ 81}

Putting value of x .y = 12 in above equation, we get;

\mathtt{x^{2} +y^{2} +2xy\ =\ 81}\\\ \\ \mathtt{x^{2} +y^{2} \ +\ 2.12\ =\ 81}\\\ \\ \mathtt{x^{2} +y^{2} \ +\ 24\ =\ 81}\\\ \\ \mathtt{x^{2} +y^{2} \ =\ 81-24}\\\ \\ \mathtt{x^{2} +y^{2} \ =\ 57}

Hence, the value of \mathtt{x^{2} +y^{2}} is 57

Example 05
What can be added in expression \mathtt{x^{2} +9y^{2}} to make it a perfect square.

Solution
Rewriting the given expression as;

\mathtt{\Longrightarrow \ x^{2} +9y^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +( 3y)^{2}}

Add 6xy to make expression in form of \mathtt{a^{2} +b^{2} +2ab}

On adding 6xy we get;

\mathtt{\Longrightarrow \ x^{2} +( 3y)^{2} +6xy\ }

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

Hence, the expression can be written as;

\mathtt{\Longrightarrow \ x^{2} +( 3y)^{2} +6xy\ }\\\ \\ \mathtt{\Longrightarrow \ ( x+3y)^{2}}

Hence, the term \mathtt{\ ( x+3y)^{2}} is a perfect square.

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