Question

Expand the following using identities: ${\left( {3x - 4y} \right)^2}$.

Answer 1

Hint: Here, in the given question, we are given an algebraic expression and we need to expand ${\left( {3x - 4y} \right)^2}$ using algebraic identities. The algebraic equations which are valid for all values of variables in them are called algebraic identities. An expression containing variables, numbers, and operation symbols is called an algebraic expression. To expand the given expression we will use ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ identity and hence on expanding the given expression, we can get our required answer.
Formula used: ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

Complete step-by-step answer:
As the given term for the expansion is, ${\left( {3x - 4y} \right)^2}$
Now, we will use formula for the expansion as ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
Hence, on expanding the algebraic expansion, we get
$ \Rightarrow {\left( {3x - 4y} \right)^2} = {\left( {3x} \right)^2} - 2\left( {3x} \right)\left( {4y} \right) + {\left( {4y} \right)^2}$
Now on expanding the terms of bracket inside R.H.S, we get
$ \Rightarrow {\left( {3x - 4y} \right)^2} = 9{x^2} - 24xy + 16{y^2}$
Hence, above is our required expanded form for the given term.
So, the correct answer is “$ 9{x^2} - 24xy + 16{y^2}$”.

Note: Remember that ${\left( {3x} \right)^2}$ is nothing but $3x \times 3x$. We have written $x \times x = {x^2}$ because according to product rule, ${a^m} \times {a^n} = {a^{m + n}}$, the product of multiplication of exponents with the same base is equal to the sum of their powers with same base. Similarly, we wrote $y \times y = {y^2}$. In order to solve questions of this type the key is to identify the nature of the given question and use the formula that fits and use it accordingly.