Question

Evaluate the following using identities:
 $ \left( {1.5{x^2} - 0.3{y^2}} \right)\left( {1.5{x^2} + 0.3{y^2}} \right). $

Answer 1

Hint: The algebraic equations which are valid for every value of variables in them are called algebraic identities.
They are also used for the factorization of polynomials.
We are using such identity in this question.
 $ \Rightarrow $ \[{a^2}\;-{\text{ }}{b^2} = {\text{ }}\left( {a{\text{ }} + {\text{ }}b} \right)\left( {a{\text{ }}-{\text{ }}b} \right)\]._ _ _ _ _ _ _ _ _ _ $ \left( 1 \right) $

Complete step-by-step answer:
The equation in the question is,
 $ \Rightarrow \left( {1.5{x^2} - 0.3{y^2}} \right)\left( {1.5{x^2} + 0.3{y^2}} \right) $
So, here we can use the equation $ \left( 1 \right) $ identity.
\[{a^2}\;-{\text{ }}{b^2} = {\text{ }}\left( {a{\text{ }} + {\text{ }}b} \right)\left( {a{\text{ }}-{\text{ }}b} \right)\]
 $ a = 1.5{x^2},b = 0.3{y^2} $
By using identity,
 $
   \Rightarrow {\left( {1.5{x^2}} \right)^2} - {\left( {0.3{y^2}} \right)^2} \\
   \Rightarrow 2.25{x^4} - 0.09{y^{{4^{}}}} \;
  $
Hence, using identities the solution is $ 2.25{x^4} - 0.09{y^{{4^{}}}} $ .
So, the correct answer is “ $ 2.25{x^4} - 0.09{y^{{4^{}}}} $ ”.

Note: $ \Rightarrow $ The algebraic identities are verified using the substitution method. In this method, substitute the values for the variables and perform the arithmetic operation.
 $
   \Rightarrow {\left( {a{\text{ }} + {\text{ }}b} \right)^2}\; = {\text{ }}{a^2}\; + {\text{ }}2ab{\text{ }} + {\text{ }}{b^2} \\
   \Rightarrow {\left( {a{\text{ }}-{\text{ }}b} \right)^2}\; = {\text{ }}{a^2}\;-{\text{ }}2ab{\text{ }} + {\text{ }}{b^2} \\
   \Rightarrow \left( {x{\text{ }} + {\text{ }}a} \right)\left( {x{\text{ }} + {\text{ }}b} \right){\text{ }} = {\text{ }}{x^2}\; + {\text{ }}\left( {a{\text{ }} + {\text{ }}b} \right){\text{ }}x{\text{ }} + {\text{ }}ab \\
\Rightarrow {\;{a^3}\; + {\text{ }}{b^3}\; + {\text{ }}{c^{3\;}}-{\text{ }}3abc\; = {\text{ }}\left( {a{\text{ }} + {\text{ }}b{\text{ }} + {\text{ }}c} \right)\left( {{a^2}\; + {\text{ }}{b^2}\; + {\text{ }}{c^2}\;-{\text{ }}ab{\text{ }}-{\text{ }}bc{\text{ }}-{\text{ }}ca} \right)} \;
$
 $ \Rightarrow $ Thus, the expression value can change if the variable values are changed. But algebraic identity is equality which is true for all the values of the variables.