Question

Factorize the expression: ${x^2} - \dfrac{{{y^2}}}{{100}}$.

Answer 1

Hint: In the given question, we have to factorize the expression given to us in the problem itself. We can do so with the help of an algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$. The algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ is used to evaluate the difference between the squares of two terms.

Complete step-by-step answer:
Given question requires us to simplify and factorize the expression provided to us in the problem as ${x^2} - \dfrac{{{y^2}}}{{100}}$.
So. let the given expression be S.
Then, $S = {x^2} - \dfrac{{{y^2}}}{{100}}$
The given expression does not have any middle term. Hence, the given can be factored using the algebraic identities. So, we can use algebraic identity $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$ to factorise the given algebraic expression.
In order to use the algebraic identity, we should convert the given expression into a form that would resemble the algebraic identity. Now, we know that $100$ is a square of $10$. So, we get,
$ \Rightarrow S = {\left( x \right)^2} - {\left( {\dfrac{y}{{10}}} \right)^2}$
Now, using the identity \[\left( {{a^2} - {b^2}} \right) = (a + b)(a - b)\] in the right side, we get,
$ \Rightarrow S = \left( {x - \dfrac{y}{{10}}} \right)\left( {x + \dfrac{y}{{10}}} \right)$
Hence, there are two factors of the algebraic expression provided to us in the problem.
So, the factored form of the given expression ${x^2} - \dfrac{{{y^2}}}{{100}}$ is $\left( {x - \dfrac{y}{{10}}} \right)\left( {x + \dfrac{y}{{10}}} \right)$.

Note: Similar to quadratic equations, quadratic expressions can also be solved using the factorisation method. We should have a strong grip over the applications of the algebraic identities in order to attempt such questions with ease. Care should be taken while using these algebraic identities calculations as it can alter the answer to the given problem. Verify the calculations once so as to be sure of the final answer.