Question

Factorize \[4{p^2} - 9{q^2}\].

Answer 1

Hint: ere we will simplify the expression by writing the constant term as the square of some number. Then we will use the suitable algebraic properties to expand the formed expression in terms of the factors. Factors are the smallest numbers with which the given number is divisible and when factors are multiplied will give the original number.

Complete step by step solution:
The given expression is \[4{p^2} - 9{q^2}\].
First, we will write the constant term of the expression in terms of the squares of a number. Therefore, we get
\[ \Rightarrow 4{p^2} - 9{q^2} = {2^2}{p^2} - {3^2}{q^2}\]
We can combine the constant terms and the variable to write in a common square. Therefore, we can write the above equation as
\[ \Rightarrow 4{p^2} - 9{q^2} = {\left( {2p} \right)^2} - {\left( {3q} \right)^2}\]
Now we will use the basic algebraic identity i.e. \[{a^2} - {b^2} = (a - b)(a + b)\] to expand this equation in the form of its factors. Therefore by expanding the equation, we get
\[ \Rightarrow 4{p^2} - 9{q^2} = \left( {2p - 3q} \right)\left( {2p + 3q} \right)\]

Hence, the factor of the given expression \[4{{p}^{2}}-9{{q}^{2}}\] is \[\left( {2p - 3q} \right),\left( {2p + 3q} \right)\].

Note:
Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number. In order to solve this question we have to use the basic algebraic identities to find the different factors of the equation. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. Here, we can make a mistake by leaving the answer at \[{\left( {2p} \right)^2} - {\left( {3q} \right)^2}\]. This is not the factored form of the given expression, rather it is a simplified form of the expression.