Square of Trinomial

In this chapter we will learn formula for square of trinomial along with solved examples at the end.

Trinomial Square Formula

If a , b & c are the given entities then square of sum of a + b + c is given by following formula.

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

Hence, square of sum (a + b + c) is equal to square of individual terms and addition of 2ab+ 2bc+ 2ca.

Deriving Square of Trinomial Formula

The given expression is: \mathtt{(a+b+c)^{2}}

Let a + b = x

Rewriting the equation;

\mathtt{\Longrightarrow \ ( x\ +\ c)^{2}}

Now using the square of sum formula;

\mathtt{\Longrightarrow \ ( x\ +\ c)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ \ x^{2} +2xc+c^{2}}


Putting the value of x = a + b in equation.

\mathtt{\Longrightarrow \ ( a+b)^{2} +2( a+b) c\ +c^{2}}


Again using sum of square formula for \mathtt{( a+b)^{2}}

\mathtt{\Longrightarrow \ a^{2} +b^{2} +2ab\ +2( a+b) c\ +c^{2}}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +b^{2} +2ab\ ++2ac+2bc+c^{2}}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +b^{2} +c^{2} +2ab+2bc+2ca}


Hence we get the formula;
\mathtt{( a+b+c)^{2} =\ a^{2} +b^{2} +c^{2} +2ab+2bc+2ca}

Proof of Square of Trinomial formula

Consider the number \mathtt{( 3+4+2)^{2}}

Finding value of number using simple calculation.

\mathtt{\Longrightarrow ( 3+4+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 9)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 81}

Hence, the value of expression is 81.



Now find the value using Square of Trinomial Formula

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

Putting the values;

\mathtt{\Longrightarrow \ ( 3+4+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 3^{2} +4^{2} +2^{2} +2.3.4+2.4.2+2.2.3}\\\ \\ \mathtt{\Longrightarrow \ 9\ +\ 16\ +\ 4\ +\ 24+\ 16\ +\ 12}\\\ \\ \mathtt{\Longrightarrow \ 81}

Using the formula we get the same value 81.

Hence the above formula is valid.

Square of Trinomial – Solved problems

Example 01
Expand \mathtt{( 2x+4y+5z)^{2}}

Solution
The expression is in form of square of trinomial.

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

Putting the values;

\mathtt{\Longrightarrow \ ( 2x+4y+5z)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 2x)^{2} +( 4y)^{2} +( 5z)^{2} + 2( 2x)( 4y) +2( 4y)( 5z) +2( 5z)( 2x)}\\\ \\ \mathtt{\Longrightarrow \ 4x^{2} +\ 16y^{2} +25z^{2} +16xy+40yz+20zx}

Hence, the above expression is the expanded form of given trinomial square.

Example 02
Expand \mathtt{( x-7y+2z)^{2}}

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

Putting the values;
\mathtt{\Longrightarrow \ ( x-7y+2z)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( x)^{2} +( -7y)^{2} +( 2z)^{2} +2( x)( -7y) +2( -7y)( 2z) +2( 2z)( x)}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +\ 49y^{2} +4z^{2} -14xy-28yz+4zx}

Hence the above expression is expanded form of trinomial square.

Example 03
Expand \mathtt{( 5x-y-3z)^{2}}

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

Putting the values;
\mathtt{\Longrightarrow \ ( 5x-y-3z)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 5x)^{2} +( -y)^{2} +( -3z)^{2} +2( 5x)( -y) +2( -y)( -3z) +2( -3z)( 5x)}\\\ \\ \mathtt{\Longrightarrow \ 25x^{2} +\ y^{2} +9z^{2} -10xy+6yz-30zx}

Hence, the above expression is the expanded term.

Example 04
Expand \mathtt{\ ( -6x-10y-3z)^{2}}

Solution
Using the square of trinomial formula;
\mathtt{( a+b+c)^{2} =\ a^{2} +b^{2} +c^{2} +2ab+2bc+2ca}

Putting the values;

\mathtt{\Longrightarrow \ ( -6x-10y-3z)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( -6x)^{2} +( -10y)^{2} +( -3z)^{2} +2( -6x)( -10y) +2( -10y)( -3z) +2( -3z)( -6x)}\\\ \\ \mathtt{\Longrightarrow \ 6x^{2} +100y^{2} +9z^{2} +120xy+60yz+36zx}

Hence, the above expression is the solution.