Question

How do you simplify this equation $\left( 3x+5y \right)\left( 3x-5y \right)$

Answer 1

Hint: Now we want to simplify the given expression. To do so we will first use distributive property which says $c\left( a+b \right)=ca+cb$ . Then we will rearrange the terms using commutative property. Now again we will use the distributive property and simplify the equation. Hence we get the required expression.

Complete step by step solution:
Now to expand the term we will first understand properties of real numbers.
First let us understand commutative property.
Commutative property for addition is given as $a+b=b+a$ similarly we commutative property for multiplication as $ab=ba$ .
Now associativity for addition is defined $a+\left( b+c \right)=\left( a+b \right)+c$ and the associativity for multiplication defined as $\left( a.b \right).c=a.\left( b.c \right)$ .
Now let us understand the distributive property.
Distributive property is defined as $a.\left( b+c \right)=a.b+a.c$ .
Now consider the given expression $\left( 3x+5y \right)\left( 3x-5y \right)$
Now in the given expression we have multiplication of two terms. First we will open one bracket using distributive property. Now according to distributive property we have $a.\left( b+c \right)=a.b+a.c$ So let $a=3x+5y$, $b=3x$ and $c=-5y$. Hence we get the expansion as,
$\Rightarrow \left( 3x+5y \right)\left( 3x \right)+\left( 3x+5y \right)\left( -5y \right)$
Now using commutative property of multiplication we have,
$\Rightarrow \left( 3x \right)\left( 3x+5y \right)+\left( -5y \right)\left( 3x+5y \right)$
Now again using distributive property we get,
$\Rightarrow \left( 3x \right)\left( 3x \right)+\left( 3x \right)\left( 5y \right)+\left( -5y \right)\left( 3x \right)+\left( -5y \right)\left( 5y \right)$
Now we know that $\left( 3x \right)\left( 3x \right)=9{{x}^{2}}$ , $\left( 3x \right)\left( 5y \right)=15xy$ and $\left( -5y \right)\left( 5y \right)=25{{y}^{2}}$
Hence on simplifying the above equation using the obtained values we get,
$\Rightarrow 9{{x}^{2}}+15xy-15xy-25{{y}^{2}}$
Now we know that 15xy – 15xy = 0, hence using this we get the equation as,
$\Rightarrow 9{{x}^{2}}-25{{y}^{2}}$
Hence the simplified expression of the given expression is $9{{x}^{2}}-25{{y}^{2}}$

Note:
Now note that the given equation is in the form of $\left( a+b \right)\left( a-b \right)$ . now we know that the formula for $\left( {{a}^{2}}-{{b}^{2}} \right)$ is given by $\left( a-b \right)\left( a+b \right)$ Hence using this formula we can easily expand the term. Hence we get $\left( 3x+5y \right)\left( 3x-5y \right)={{\left( 3x \right)}^{2}}-{{\left( 5y \right)}^{2}}$ . Now on taking the square of term we will get the required solution.