Question

Use suitable identity to solve (6x-7)(6x+7).

Answer 1

Hint: To solve the algebraic expression, suitable identities will be used and by applying these suitable identities we can solve our algebraic equations very easily. The distributive property of multiplication is used in proving the identity, according to which if we change the order of multiplication of two variables then the result will not be changed.

Complete step by step answer:
we have to solve the equation \[(6x-7)(6x+7)\].
We know that, according to the identity.
\[(a-b)(a+b)={{a}^{2}}-{{b}^{2}}\]
Where ‘a’ and ‘b’ are two numbers or variables in the equation. In the above identity, we have two binomials. One binomial is formed by adding the two numbers or variables and the other binomial is obtained by subtracting the two variables and numbers. When these two binomials are multiplied then we get a special type of binomial in which we have a difference of the square of both the variables.
Now we will prove the above identity using the suitable method and that method is also known as the algebraic method.
In the identity, we have two binomials \[(a+b)\] and \[(a-b)\]. These binomials are algebraic expressions so can be multiplied using algebraic expressions. Now we will multiply both the binomials.
The first term of the first binomial will be multiplied by the second binomial and after that, the second term of the first binomial will be multiplied by the second binomial.
\[(a-b)(a+b)=a(a+b)-b(a+b)\].
Now we will multiply the above binomial expression with the help of distributive property which is as follows.
\[(a-b)(a+b)={{a}^{2}}+ab-ba-{{b}^{2}}\]
according to the commutative property of the multiplication, the value of \[ab\] will be equal to the value of \[ba\]. After putting these values in the above equation, we get the following results.
\[(a-b)(a+b)={{a}^{2}}-ab+ab-{{b}^{2}}\]
 In the above expression, we have two \[ab\]with opposite signs, so they both will cancel out each other. So the following expression will be obtained.
\[(a-b)(a+b)={{a}^{2}}-{{b}^{2}}\]
Hence the identity is proved.
On applying this identity in the above equation, we get.
In the above equation.
\[\begin{align}
  & a=6x \\
 & b=7 \\
\end{align}\]
\[\begin{align}
  & (6x-7)(6x+7)={{(6x)}^{2}}-{{(7)}^{2}} \\
 & \Rightarrow (6x-7)(6x+7)=36{{x}^{2}}-49 \\
\end{align}\]

Note:
 To solve the algebraic expression, various properties are used. The first one is commutative property, according to this if we change the order of terms in addition and subtraction then the result will not be changed. Distributive law can be applied to both addition and multiplication at the same time.