Question

How do you find the product \[{{\left( 2x+7y \right)}^{2}}\]?

Answer 1

Hint: In this question, we have to find the product of an equation or square of that equation which is in the linear form. We will solve this question using the foil method. The foil method says that if we multiply (a+b) and (c+d), then (a+b)(c+d) will be equal to (ac+ad+bc+bd). After solving this question, we will see the alternate method of solving this question.

Complete step by step answer:
Let us solve this question.
We have to find the product of \[{{\left( 2x+7y \right)}^{2}}\].
That means we have to find the square of (2x+7y) or to multiply (2x+7y) two times.
So, first, we should know that square of x is equal to \[x\times x={{x}^{2}}\]
And a square of y is equal to \[y\times y={{y}^{2}}\]
Let us now find the product of \[{{\left( 2x+7y \right)}^{2}}\].
A square of x can be written as \[x\times x={{x}^{2}}\].
So, we can say that \[{{\left( 2x+7y \right)}^{2}}\] can be written as \[\left( 2x+7y \right)\times \left( 2x+7y \right)\].
Now, we will find the multiplication.
By using foil method here, we can write the equation as
\[\left( 2x+7y \right)\times \left( 2x+7y \right)=2x\left( 2x+7y \right)+7y\left( 2x+7y \right)\]
Which is also can be written as
\[\Rightarrow \left( 2x+7y \right)\left( 2x+7y \right)=2x\times 2x+2x\times 7y+7y\times 2x+7y\times 7y\]
As \[2x\times 2x\] is equal to \[4{{x}^{2}}\] ,
\[7y\times 7y\] is equal to \[49{{y}^{2}}\], and
\[2x\times 7y=7y\times 2x=14xy\]
Using these 3, we can write
\[\Rightarrow \left( 2x+7y \right)\left( 2x+7y \right)=4{{x}^{2}}+14xy+14xy+49{{y}^{2}}\]
\[\Rightarrow \left( 2x+7y \right)\left( 2x+7y \right)=4{{x}^{2}}+28xy+49{{y}^{2}}\]
Hence, the product of \[{{\left( 2x+7y \right)}^{2}}\] is \[4{{x}^{2}}+28xy+49{{y}^{2}}\].

Note:
We have an alternate method to solve this question.
Let us do the question from that method.
Whenever we have to find the square of an equation which is in the form \[\left( a+b \right)\].
Then we use the formula of the square of (a+b).
That formula is \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
Hence, using the formula, we can write
\[{{\left( 2x+7y \right)}^{2}}={{\left( 2x \right)}^{2}}+2\times 2x\times 7y+{{\left( 7y \right)}^{2}}=4{{x}^{2}}+28xy+49{{y}^{2}}\]
So, from this method also, we get the same value. So, it is very useful. One can remember this method to solve this type of question.